OpenMath Content Dictionary: transc1

Canonical URL:

http://www.openmath.org/cd/transc1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

transc1.ocd

CD as XML Encoded OpenMath:

transc1.omcd

Defines:

arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, exp, ln, log, sec, sech, sin, sinh, tan, tanh

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2006-03-30

Status:

official

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

This CD holds the definitions of many transcendental functions. They are defined as in Abromowitz and Stegun (ninth printing on), with precise reductions to logs in the case of inverse functions.

Note that, if signed zeros are supported, some strict inequalities have to become weak . It is intended to be `compatible' with the MathML elements denoting trancendental functions. Some additional functions are in the CD transc2.

---

log

Role:

application

Description:

This symbol represents a binary log function; the first argument is the base, to which the second argument is log'ed. It is defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1

Commented Mathematical property (CMP):

a^b = c implies log_a c = b

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="power"/>
          <OMV name="a"/>
          <OMV name="b"/>
        </OMA>
        <OMV name="c"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="transc1" name="log"/>
          <OMV name="a"/>
          <OMV name="c"/>
        </OMA>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><ci>b</ci></apply>
   <ci>c</ci>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="transc1">log</csymbol><ci>a</ci><ci>c</ci></apply>
   <ci>b</ci>
  </apply>
 </apply>
</math>

Prefix

implies (eq (power ( a, b) , c) , eq (log ( a, c) , b) )

Popcorn

$a ^ $b = $c ==> log($a, $c) = $b

Rendered Presentation MathML

$$a^b=c\Rightarrow {\log }_a\left(c\right)=b$$

Example:

log 100 to base 10 (which is 2).

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="transc1" name="log"/>
    <OMF dec="10"/>
    <OMF dec="100"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="transc1">log</csymbol>
  <cn type="real">10</cn>
  <cn type="real">100</cn>
 </apply>
</math>

Prefix

log ( 10 , 100 )

Popcorn

log(10, 100)

Rendered Presentation MathML

$${\log }_{10}\left(100\right)$$

Signatures:

sts

---

[Next: ln] [Last: arccoth] [Top]

---

ln

Role:

application

Description:

This symbol represents the ln function (natural logarithm) as described in Abramowitz and Stegun, section 4.1. It takes one argument. Note the description in the CMP/FMP of the branch cut. If signed zeros are in use, the inequality needs to be non-strict.

Commented Mathematical property (CMP):

-pi < Im ln x <= pi

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS name="and" cd="logic1"/>
      <OMA>
        <OMS name="lt" cd="relation1"/>
        <OMA>
          <OMS name="unary_minus" cd="arith1"/>
          <OMS name="pi" cd="nums1"/>
        </OMA>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMV name="x"/>
          </OMA>
        </OMA>
      </OMA>
      <OMA>
        <OMS name="leq" cd="relation1"/>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMV name="x"/>
          </OMA>
        </OMA>
        <OMS name="pi" cd="nums1"/>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="relation1">lt</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">pi</csymbol></apply>
   <apply><csymbol cd="complex1">imaginary</csymbol>
    <apply><csymbol cd="transc1">ln</csymbol><ci>x</ci></apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">leq</csymbol>
   <apply><csymbol cd="complex1">imaginary</csymbol>
    <apply><csymbol cd="transc1">ln</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="nums1">pi</csymbol>
  </apply>
 </apply>
</math>

Prefix

and (lt (unary_minus (pi) , imaginary (ln ( x) ) ) , leq (imaginary (ln ( x) ) , pi) )

Popcorn

-(nums1.pi) < complex1.imaginary(ln($x)) and complex1.imaginary(ln($x)) <= nums1.pi

Rendered Presentation MathML

$$-\pi <\mathrm{imaginary}\left(\ln \left(x\right)\right)\wedge \mathrm{imaginary}\left(\ln \left(x\right)\right)\le \pi$$

Example:

ln 1 (which is 0).

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="transc1" name="ln"/>
    <OMF dec="1"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="transc1">ln</csymbol><cn type="real">1</cn></apply></math>

Prefix

ln ( 1 )

Popcorn

ln(1)

Rendered Presentation MathML

$$\ln \left(1\right)$$

Signatures:

sts

---

[Next: exp] [Previous: log] [Top]

---

exp

Role:

application

Description:

This symbol represents the exponentiation function as described in Abramowitz and Stegun, section 4.2. It takes one argument.

Commented Mathematical property (CMP):

for all k if k is an integer then e^(z+2*pi*k*i)=e^z

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="k"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="k"/>
      <OMS cd="setname1" name="Z"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="exp"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="z"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMI>2</OMI>
	    <OMS cd="nums1" name="pi"/>
	    <OMV name="k"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="exp"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>k</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>k</ci><csymbol cd="setname1">Z</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol>
      <ci>z</ci>
      <apply><csymbol cd="arith1">times</csymbol>
       <cn type="integer">2</cn>
       <csymbol cd="nums1">pi</csymbol>
       <ci>k</ci>
       <csymbol cd="nums1">i</csymbol>
      </apply>
     </apply>
    </apply>
    <apply><csymbol cd="transc1">exp</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ k ] . (implies (in ( k, Z) , eq (exp (plus ( z, times (2, pi, k, i) ) ) , exp ( z) ) ) )

Popcorn

quant1.forall[$k -> set1.in($k, setname1.Z) ==> exp($z + 2 * nums1.pi * $k * nums1.i) = exp($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}k.k\in \mathbb{Z}\Rightarrow \exp \left(z+2\pi ki\right)=\exp \left(z\right)$$

Signatures:

sts

---

[Next: sin] [Previous: ln] [Top]

---

sin

Role:

application

Description:

This symbol represents the sin function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

sin(x) = (exp(ix)-exp(-ix))/2i

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS name="sin" cd="transc1"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS name="divide" cd="arith1"/>
        <OMA>
          <OMS name="minus" cd="arith1"/>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="x"/>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMA>
                <OMS name="unary_minus" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
              </OMA>
              <OMV name="x"/>
            </OMA>
          </OMA>
        </OMA>
        <OMA>
          <OMS name="times" cd="arith1"/>
          <OMI>2</OMI>
          <OMS name="i" cd="nums1"/>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>x</ci></apply>
    </apply>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
      <ci>x</ci>
     </apply>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <cn type="integer">2</cn>
    <csymbol cd="nums1">i</csymbol>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (sin ( x) , divide (minus (exp (times (i, x) ) , exp (times (unary_minus (i) , x) ) ) , times (2, i) ) )

Popcorn

sin($x) = (exp(nums1.i * $x) - exp( -(nums1.i) * $x)) / (2 * nums1.i)

Rendered Presentation MathML

$$\sin \left(x\right)=\frac{\exp \left(ix\right)-\exp \left(-ix\right)}{2i}$$

Commented Mathematical property (CMP):

sin(A + B) = sin A cos B + cos A sin B

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
        <OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="A"/>
    	  <OMV name="B"/>
        </OMA>
      </OMA>
      <OMA>
	<OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="A"/>
	  </OMA>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="B"/>
	  </OMA>
	</OMA>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="A"/>
	  </OMA>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="B"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sin</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="transc1">sin</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="transc1">cos</csymbol><ci>B</ci></apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="transc1">cos</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="transc1">sin</csymbol><ci>B</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (sin (plus ( A, B) ) , plus (times (sin ( A) , cos ( B) ) , times (cos ( A) , sin ( B) ) ) )

Popcorn

sin($A + $B) = sin($A) * cos($B) + cos($A) * sin($B)

Rendered Presentation MathML

$$\sin \left(A+B\right)=\sin \left(A\right)\cos \left(B\right)+\cos \left(A\right)\sin \left(B\right)$$

Commented Mathematical property (CMP):

sin A = - sin(-A)

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="transc1" name="sin"/>
	  <OMA>
	    <OMS cd="arith1" name="unary_minus"/>
	    <OMV name="A"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sin</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">unary_minus</csymbol>
   <apply><csymbol cd="transc1">sin</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol><ci>A</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (sin ( A) , unary_minus (sin (unary_minus ( A) ) ) )

Popcorn

sin($A) = -(sin( -($A)))

Rendered Presentation MathML

$$\sin \left(A\right)=-\sin \left(-A\right)$$

Signatures:

sts

---

[Next: cos] [Previous: exp] [Top]

---

cos

Role:

application

Description:

This symbol represents the cos function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

cos(x) = (exp(ix)+exp(-ix))/2

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS name="cos" cd="transc1"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS name="divide" cd="arith1"/>
        <OMA>
          <OMS name="plus" cd="arith1"/>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="x"/>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMA>
                <OMS name="unary_minus" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
              </OMA>
              <OMV name="x"/>
            </OMA>
          </OMA>
        </OMA>
        <OMI>2</OMI>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">cos</csymbol><ci>x</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>x</ci></apply>
    </apply>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
      <ci>x</ci>
     </apply>
    </apply>
   </apply>
   <cn type="integer">2</cn>
  </apply>
 </apply>
</math>

Prefix

eq (cos ( x) , divide (plus (exp (times (i, x) ) , exp (times (unary_minus (i) , x) ) ) , 2) )

Popcorn

cos($x) = (exp(nums1.i * $x) + exp( -(nums1.i) * $x)) / 2

Rendered Presentation MathML

$$\cos \left(x\right)=\frac{\exp \left(ix\right)+\exp \left(-ix\right)}{2}$$

Commented Mathematical property (CMP):

cos 2A = cos^2 A - sin^2 A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMI> 2 </OMI>
	<OMV name="A"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMA>
	  <OMS cd="transc1" name="cos"/>
	  <OMV name="A"/>
	</OMA>
	<OMI> 2 </OMI>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMA>
	  <OMS cd="transc1" name="sin"/>
	  <OMV name="A"/>
	</OMA>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">cos</csymbol>
   <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>A</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="arith1">power</csymbol>
    <apply><csymbol cd="transc1">cos</csymbol><ci>A</ci></apply>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="arith1">power</csymbol>
    <apply><csymbol cd="transc1">sin</csymbol><ci>A</ci></apply>
    <cn type="integer">2</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (cos (times ( 2 , A) ) , minus (power (cos ( A) , 2 ) , power (sin ( A) , 2 ) ) )

Popcorn

cos(2 * $A) = cos($A) ^ 2 - sin($A) ^ 2

Rendered Presentation MathML

$$\cos \left(2A\right)={\cos \left(A\right)}^2-{\sin \left(A\right)}^2$$

Commented Mathematical property (CMP):

cos A = cos(-A)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMV name="A"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">cos</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="transc1">cos</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (cos ( A) , cos (unary_minus ( A) ) )

Popcorn

cos($A) = cos( -($A))

Rendered Presentation MathML

$$\cos \left(A\right)=\cos \left(-A\right)$$

Signatures:

sts

---

[Next: tan] [Previous: sin] [Top]

---

tan

Role:

application

Description:

This symbol represents the tan function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

tan A = sin A / cos A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="tan"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMA>
      <OMS cd="transc1" name="sin"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">tan</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="transc1">sin</csymbol><ci>A</ci></apply>
   <apply><csymbol cd="transc1">cos</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (tan ( A) , divide (sin ( A) , cos ( A) ) )

Popcorn

tan($A) = sin($A) / cos($A)

Rendered Presentation MathML

$$\tan \left(A\right)=\frac{\sin \left(A\right)}{\cos \left(A\right)}$$

Signatures:

sts

---

[Next: sec] [Previous: cos] [Top]

---

sec

Role:

application

Description:

This symbol represents the sec function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

sec A = 1/cos A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sec"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sec</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">cos</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (sec ( A) , divide (one, cos ( A) ) )

Popcorn

sec($A) = alg1.one / cos($A)

Rendered Presentation MathML

$$\sec \left(A\right)=\frac{1}{\cos \left(A\right)}$$

Signatures:

sts

---

[Next: csc] [Previous: tan] [Top]

---

csc

Role:

application

Description:

This symbol represents the csc function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

csc A = 1/sin A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="csc"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="sin"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">csc</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">sin</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (csc ( A) , divide (one, sin ( A) ) )

Popcorn

csc($A) = alg1.one / sin($A)

Rendered Presentation MathML

$$\csc \left(A\right)=\frac{1}{\sin \left(A\right)}$$

Signatures:

sts

---

[Next: cot] [Previous: sec] [Top]

---

cot

Role:

application

Description:

This symbol represents the cot function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):

cot A = 1/tan A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="cot"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="tan"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">cot</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">tan</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (cot ( A) , divide (one, tan ( A) ) )

Popcorn

cot($A) = alg1.one / tan($A)

Rendered Presentation MathML

$$\cot \left(A\right)=\frac{1}{\tan \left(A\right)}$$

Signatures:

sts

---

[Next: sinh] [Previous: csc] [Top]

---

sinh

Role:

application

Description:

This symbol represents the sinh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

sinh A = 1/2 * (e^A - e^(-A))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sinh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="nums1" name="rational"/>
      <OMS cd="alg1" name="one"/>
      <OMI> 2 </OMI>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMA>
	  <OMS cd="arith1" name="unary_minus"/>
	  <OMV name="A"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sinh</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="nums1">rational</csymbol>
    <csymbol cd="alg1">one</csymbol>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="arith1">power</csymbol><csymbol cd="nums1">e</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="arith1">power</csymbol>
     <csymbol cd="nums1">e</csymbol>
     <apply><csymbol cd="arith1">unary_minus</csymbol><ci>A</ci></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (sinh ( A) , times (rational (one, 2 ) , minus (power (e, A) , power (e, unary_minus ( A) ) ) ) )

Popcorn

sinh($A) = alg1.one // 2 * (nums1.e ^ $A - nums1.e ^ -($A))

Rendered Presentation MathML

$$\sinh \left(A\right)=\frac{1}{2}(e^A-e^{\left(-A\right)})$$

Signatures:

sts

---

[Next: cosh] [Previous: cot] [Top]

---

cosh

Role:

application

Description:

This symbol represents the cosh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

cosh A = 1/2 * (e^A + e^(-A))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="cosh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="nums1" name="rational"/>
      <OMS cd="alg1" name="one"/>
      <OMI> 2 </OMI>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMA>
	  <OMS cd="arith1" name="unary_minus"/>
	  <OMV name="A"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">cosh</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="nums1">rational</csymbol>
    <csymbol cd="alg1">one</csymbol>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="arith1">power</csymbol><csymbol cd="nums1">e</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="arith1">power</csymbol>
     <csymbol cd="nums1">e</csymbol>
     <apply><csymbol cd="arith1">unary_minus</csymbol><ci>A</ci></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (cosh ( A) , times (rational (one, 2 ) , plus (power (e, A) , power (e, unary_minus ( A) ) ) ) )

Popcorn

cosh($A) = alg1.one // 2 * (nums1.e ^ $A + nums1.e ^ -($A))

Rendered Presentation MathML

$$\cosh \left(A\right)=\frac{1}{2}(e^A+e^{\left(-A\right)})$$

Signatures:

sts

---

[Next: tanh] [Previous: sinh] [Top]

---

tanh

Role:

application

Description:

This symbol represents the tanh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

tanh A = sinh A / cosh A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="tanh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMA>
      <OMS cd="transc1" name="sinh"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cosh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">tanh</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="transc1">sinh</csymbol><ci>A</ci></apply>
   <apply><csymbol cd="transc1">cosh</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (tanh ( A) , divide (sinh ( A) , cosh ( A) ) )

Popcorn

tanh($A) = sinh($A) / cosh($A)

Rendered Presentation MathML

$$\tanh \left(A\right)=\frac{\sinh \left(A\right)}{\cosh \left(A\right)}$$

Signatures:

sts

---

[Next: sech] [Previous: cosh] [Top]

---

sech

Role:

application

Description:

This symbol represents the sech function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

sech A = 1/cosh A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sech"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="cosh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">sech</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">cosh</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (sech ( A) , divide (one, cosh ( A) ) )

Popcorn

sech($A) = alg1.one / cosh($A)

Rendered Presentation MathML

$$\mathrm{sech}\left(A\right)=\frac{1}{\cosh \left(A\right)}$$

Signatures:

sts

---

[Next: csch] [Previous: tanh] [Top]

---

csch

Role:

application

Description:

This symbol represents the csch function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

csch A = 1/sinh A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="csch"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="sinh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">csch</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">sinh</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (csch ( A) , divide (one, sinh ( A) ) )

Popcorn

csch($A) = alg1.one / sinh($A)

Rendered Presentation MathML

$$\mathrm{csch}\left(A\right)=\frac{1}{\sinh \left(A\right)}$$

Signatures:

sts

---

[Next: coth] [Previous: sech] [Top]

---

coth

Role:

application

Description:

This symbol represents the coth function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):

coth A = 1/tanh A

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="coth"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="tanh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">coth</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="transc1">tanh</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (coth ( A) , divide (one, tanh ( A) ) )

Popcorn

coth($A) = alg1.one / tanh($A)

Rendered Presentation MathML

$$\coth \left(A\right)=\frac{1}{\tanh \left(A\right)}$$

Signatures:

sts

---

[Next: arcsin] [Previous: csch] [Top]

---

arcsin

Role:

application

Description:

This symbol represents the arcsin function. This is the inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):

arcsin(z) = -i ln (sqrt(1-z^2)+iz)

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS name="eq" cd="relation1"/>
      <OMA>
        <OMS name="arcsin" cd="transc1"/>
        <OMV name="z"/>
      </OMA>
      <OMA>
        <OMS name="times" cd="arith1"/>
        <OMA>
          <OMS name="unary_minus" cd="arith1"/>
          <OMS name="i" cd="nums1"/>
        </OMA>
        <OMA>
          <OMS name="ln" cd="transc1"/>
          <OMA>
            <OMS name="plus" cd="arith1"/>
            <OMA>
              <OMS name="root" cd="arith1"/>
              <OMA>
                <OMS name="minus" cd="arith1"/>
                <OMS name="one" cd="alg1"/>
                <OMA>
                  <OMS name="power" cd="arith1"/>
                  <OMV name="z"/>
                  <OMI> 2 </OMI>
                </OMA>
              </OMA>
              <OMI> 2 </OMI>
            </OMA>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arcsin</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol>
       <csymbol cd="alg1">one</csymbol>
       <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
      </apply>
      <cn type="integer">2</cn>
     </apply>
     <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arcsin ( z) , times (unary_minus (i) , ln (plus (root (minus (one, power ( z, 2 ) ) , 2 ) , times (i, z) ) ) ) )

Popcorn

arcsin($z) = -(nums1.i) * ln(arith1.root(alg1.one - $z ^ 2, 2) + nums1.i * $z)

Rendered Presentation MathML

$$\arcsin \left(z\right)=-i\ln \left(\sqrt{1-z^2}+iz\right)$$

Commented Mathematical property (CMP):

x in [-(pi/2),(pi/2)] implies arcsin(sin x) = x

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_cc"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="nums1" name="pi"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="nums1" name="pi"/>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsin"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_cc</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <csymbol cd="nums1">pi</csymbol>
      <cn type="integer">2</cn>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">divide</csymbol>
     <csymbol cd="nums1">pi</csymbol>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="transc1">arcsin</csymbol>
    <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
   </apply>
   <ci>x</ci>
  </apply>
 </apply>
</math>

Prefix

implies (in ( x, interval_cc (unary_minus (divide (pi, 2 ) ) , divide (pi, 2 ) ) ) , eq (arcsin (sin ( x) ) , x) )

Popcorn

set1.in($x, interval1.interval_cc( -(nums1.pi / 2), nums1.pi / 2)) ==> arcsin(sin($x)) = $x

Rendered Presentation MathML

$$x\in [-\left(\frac{\pi }{2}\right),\frac{\pi }{2}]\Rightarrow \arcsin \left(\sin \left(x\right)\right)=x$$

Signatures:

sts

---

[Next: arccos] [Previous: coth] [Top]

---

arccos

Role:

application

Description:

This symbol represents the arccos function. This is the inverse of the cos function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):

arccos(z) = -i ln(z+i \sqrt(1-z^2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccos"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="z"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMA>
	      <OMS cd="arith1" name="root"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="z"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccos</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <ci>z</ci>
     <apply><csymbol cd="arith1">times</csymbol>
      <csymbol cd="nums1">i</csymbol>
      <apply><csymbol cd="arith1">root</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol>
        <csymbol cd="alg1">one</csymbol>
        <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
       </apply>
       <cn type="integer">2</cn>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccos ( z) , times (unary_minus (i) , ln (plus ( z, times (i, root (minus (one, power ( z, 2 ) ) , 2 ) ) ) ) ) )

Popcorn

arccos($z) = -(nums1.i) * ln($z + nums1.i * arith1.root(alg1.one - $z ^ 2, 2))

Rendered Presentation MathML

$$\arccos \left(z\right)=-i\ln \left(z+i\sqrt{1-z^2}\right)$$

Commented Mathematical property (CMP):

x in [0,pi] implies arccos(cos x) = x

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_cc"/>
      <OMS cd="alg1" name="zero"/>
      <OMS cd="nums1" name="pi"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccos"/>
      <OMA>
        <OMS cd="transc1" name="cos"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_cc</csymbol>
    <csymbol cd="alg1">zero</csymbol>
    <csymbol cd="nums1">pi</csymbol>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="transc1">arccos</csymbol>
    <apply><csymbol cd="transc1">cos</csymbol><ci>x</ci></apply>
   </apply>
   <ci>x</ci>
  </apply>
 </apply>
</math>

Prefix

implies (in ( x, interval_cc (zero, pi) ) , eq (arccos (cos ( x) ) , x) )

Popcorn

set1.in($x, interval1.interval_cc(alg1.zero, nums1.pi)) ==> arccos(cos($x)) = $x

Rendered Presentation MathML

$$x\in [0,\pi ]\Rightarrow \arccos \left(\cos \left(x\right)\right)=x$$

Signatures:

sts

---

[Next: arctan] [Previous: arcsin] [Top]

---

arctan

Role:

application

Description:

This symbol represents the arctan function. This is the inverse of the tan function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):

arctan(z) = (i/2)ln((1-iz)/(1+iz))

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS name="eq" cd="relation1"/>
      <OMA>
        <OMS name="arctan" cd="transc1"/>
        <OMV name="z"/>
      </OMA>
      <OMA>
        <OMS name="times" cd="arith1"/>
        <OMA>
          <OMS name="divide" cd="arith1"/>
          <OMS name="i" cd="nums1"/>
          <OMI> 2 </OMI>
        </OMA>
        <OMA>
          <OMS name="ln" cd="transc1"/>
	  <OMA>
            <OMS name="divide" cd="arith1"/>
            <OMA>
              <OMS name="minus" cd="arith1"/>
              <OMS name="one" cd="alg1"/>
              <OMA>
                <OMS name="times" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
            <OMA>
              <OMS name="plus" cd="arith1"/>
              <OMS name="one" cd="alg1"/>
              <OMA>
                <OMS name="times" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arctan</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <csymbol cd="nums1">i</csymbol>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol>
      <csymbol cd="alg1">one</csymbol>
      <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
     </apply>
     <apply><csymbol cd="arith1">plus</csymbol>
      <csymbol cd="alg1">one</csymbol>
      <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arctan ( z) , times (divide (i, 2 ) , ln (divide (minus (one, times (i, z) ) , plus (one, times (i, z) ) ) ) ) )

Popcorn

arctan($z) = nums1.i / 2 * ln((alg1.one - nums1.i * $z) / (alg1.one + nums1.i * $z))

Rendered Presentation MathML

$$\arctan \left(z\right)=\frac{i}{2}\ln \left(\frac{1-iz}{1+iz}\right)$$

Commented Mathematical property (CMP):

x in (-(pi/2),(pi/2)) implies arctan(tan x) = x

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_oo"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="nums1" name="pi"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="nums1" name="pi"/>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arctan"/>
      <OMA>
        <OMS cd="transc1" name="tan"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_oo</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <csymbol cd="nums1">pi</csymbol>
      <cn type="integer">2</cn>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">divide</csymbol>
     <csymbol cd="nums1">pi</csymbol>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="transc1">arctan</csymbol>
    <apply><csymbol cd="transc1">tan</csymbol><ci>x</ci></apply>
   </apply>
   <ci>x</ci>
  </apply>
 </apply>
</math>

Prefix

implies (in ( x, interval_oo (unary_minus (divide (pi, 2 ) ) , divide (pi, 2 ) ) ) , eq (arctan (tan ( x) ) , x) )

Popcorn

set1.in($x, interval1.interval_oo( -(nums1.pi / 2), nums1.pi / 2)) ==> arctan(tan($x)) = $x

Rendered Presentation MathML

$$x\in (-\left(\frac{\pi }{2}\right),\frac{\pi }{2})\Rightarrow \arctan \left(\tan \left(x\right)\right)=x$$

Signatures:

sts

---

[Next: arcsec] [Previous: arccos] [Top]

---

arcsec

Role:

application

Description:

This symbol represents the arcsec function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):

arcsec(z) = -i ln(1/z + i \sqrt(1-1/z^2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsec"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMA>
	      <OMS cd="arith1" name="root"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="divide"/>
		  <OMS cd="alg1" name="one"/>
		  <OMA>
		    <OMS cd="arith1" name="power"/>
		    <OMV name="z"/>
		    <OMI> 2 </OMI>
		  </OMA>
		</OMA>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arcsec</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
     <apply><csymbol cd="arith1">times</csymbol>
      <csymbol cd="nums1">i</csymbol>
      <apply><csymbol cd="arith1">root</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol>
        <csymbol cd="alg1">one</csymbol>
        <apply><csymbol cd="arith1">divide</csymbol>
         <csymbol cd="alg1">one</csymbol>
         <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
        </apply>
       </apply>
       <cn type="integer">2</cn>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arcsec ( z) , times (unary_minus (i) , ln (plus (divide (one, z) , times (i, root (minus (one, divide (one, power ( z, 2 ) ) ) , 2 ) ) ) ) ) )

Popcorn

arcsec($z) = -(nums1.i) * ln(alg1.one / $z + nums1.i * arith1.root(alg1.one - alg1.one / $z ^ 2, 2))

Rendered Presentation MathML

$$\mathrm{arcsec}\left(z\right)=-i\ln \left(\frac{1}{z}+i\sqrt{1-\frac{1}{z^2}}\right)$$

Commented Mathematical property (CMP):

for all z | arcsec z = i * arcsech z

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="z"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsec"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="nums1" name="i"/>
      <OMA>
        <OMS cd="transc1" name="arcsech"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="transc1">arcsec</csymbol><ci>z</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <csymbol cd="nums1">i</csymbol>
    <apply><csymbol cd="transc1">arcsech</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (eq (arcsec ( z) , times (i, arcsech ( z) ) ) )

Popcorn

quant1.forall[$z -> arcsec($z) = nums1.i * arcsech($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.\mathrm{arcsec}\left(z\right)=i\mathrm{arcsech}\left(z\right)$$

Signatures:

sts

---

[Next: arccsc] [Previous: arctan] [Top]

---

arccsc

Role:

application

Description:

This symbol represents the arccsc function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):

arccsc(z) = -i ln(i/z + \sqrt(1 - 1/z^2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="nums1" name="i"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMS cd="alg1" name="one"/>
	      <OMA>
		<OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="z"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccsc</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol>
       <csymbol cd="alg1">one</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol>
        <csymbol cd="alg1">one</csymbol>
        <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
       </apply>
      </apply>
      <cn type="integer">2</cn>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccsc ( z) , times (unary_minus (i) , ln (plus (divide (i, z) , root (minus (one, divide (one, power ( z, 2 ) ) ) , 2 ) ) ) ) )

Popcorn

arccsc($z) = -(nums1.i) * ln(nums1.i / $z + arith1.root(alg1.one - alg1.one / $z ^ 2, 2))

Rendered Presentation MathML

$$\mathrm{arccsc}\left(z\right)=-i\ln \left(\frac{i}{z}+\sqrt{1-\frac{1}{z^2}}\right)$$

Commented Mathematical property (CMP):

arccsc(z) = i * arccsch(i * z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsc"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="transc1" name="arccsch"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccsc</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <csymbol cd="nums1">i</csymbol>
   <apply><csymbol cd="transc1">arccsch</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccsc ( z) , times (i, arccsch (times (i, z) ) ) )

Popcorn

arccsc($z) = nums1.i * arccsch(nums1.i * $z)

Rendered Presentation MathML

$$\mathrm{arccsc}\left(z\right)=i\mathrm{arccsch}\left(iz\right)$$

Commented Mathematical property (CMP):

arccsc(-z) = - arccsc(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsc"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="unary_minus"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccsc</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><ci>z</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">unary_minus</csymbol>
   <apply><csymbol cd="transc1">arccsc</csymbol><ci>z</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccsc (unary_minus ( z) ) , unary_minus (arccsc ( z) ) )

Popcorn

arccsc( -($z)) = -(arccsc($z))

Rendered Presentation MathML

$$\mathrm{arccsc}\left(-z\right)=-\mathrm{arccsc}\left(z\right)$$

Signatures:

sts

---

[Next: arccot] [Previous: arcsec] [Top]

---

arccot

Role:

application

Description:

This symbol represents the arccot function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):

arccot(-z) = - arccot(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccot"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="unary_minus"/>
    <OMA>
      <OMS cd="transc1" name="arccot"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccot</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><ci>z</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">unary_minus</csymbol>
   <apply><csymbol cd="transc1">arccot</csymbol><ci>z</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccot (unary_minus ( z) ) , unary_minus (arccot ( z) ) )

Popcorn

arccot( -($z)) = -(arccot($z))

Rendered Presentation MathML

$$\mathrm{arccot}\left(-z\right)=-\mathrm{arccot}\left(z\right)$$

Commented Mathematical property (CMP):

arccot(x) = (i/2) * ln ((x - i)/(x + i))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccot"/>
      <OMV name="x"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="nums1" name="i"/>
	<OMI> 2 </OMI>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
        <OMA>
          <OMS cd="arith1" name="divide"/>
	  <OMA>
	    <OMS cd="arith1" name="minus"/>
	    <OMV name="x"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMV name="x"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccot</csymbol><ci>x</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <csymbol cd="nums1">i</csymbol>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol><ci>x</ci><csymbol cd="nums1">i</csymbol></apply>
     <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><csymbol cd="nums1">i</csymbol></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccot ( x) , times (divide (i, 2 ) , ln (divide (minus ( x, i) , plus ( x, i) ) ) ) )

Popcorn

arccot($x) = nums1.i / 2 * ln(($x - nums1.i) / ($x + nums1.i))

Rendered Presentation MathML

$$\mathrm{arccot}\left(x\right)=\frac{i}{2}\ln \left(\frac{x-i}{x+i}\right)$$

Signatures:

sts

---

[Next: arcsinh] [Previous: arccsc] [Top]

---

arcsinh

Role:

application

Description:

This symbol represents the arcsinh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arcsinh z = ln(z + \sqrt(1+z^2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsinh"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="ln"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMV name="z"/>
	<OMA>
	  <OMS cd="arith1" name="root"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMV name="z"/>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arcsinh</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="transc1">ln</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <ci>z</ci>
    <apply><csymbol cd="arith1">root</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol>
      <csymbol cd="alg1">one</csymbol>
      <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
     </apply>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arcsinh ( z) , ln (plus ( z, root (plus (one, power ( z, 2 ) ) , 2 ) ) ) )

Popcorn

arcsinh($z) = ln($z + arith1.root(alg1.one + $z ^ 2, 2))

Rendered Presentation MathML

$$\mathrm{arcsinh}\left(z\right)=\ln \left(z+\sqrt{1+z^2}\right)$$

Commented Mathematical property (CMP):

arcsinh(z) = - i * arcsin(i * z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arcsinh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMS cd="nums1" name="i"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="arcsin"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arcsinh</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">arcsin</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arcsinh ( z) , times (unary_minus (i) , arcsin (times (i, z) ) ) )

Popcorn

arcsinh($z) = -(nums1.i) * arcsin(nums1.i * $z)

Rendered Presentation MathML

$$\mathrm{arcsinh}\left(z\right)=-i\arcsin \left(iz\right)$$

Signatures:

sts

---

[Next: arccosh] [Previous: arccot] [Top]

---

arccosh

Role:

application

Description:

This symbol represents the arccosh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arccosh(z) = 2*ln(\sqrt((z+1)/2) + \sqrt((z-1)/2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccosh"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="plus"/>
		<OMV name="z"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="z"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccosh</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <cn type="integer">2</cn>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">plus</csymbol><ci>z</ci><csymbol cd="alg1">one</csymbol></apply>
       <cn type="integer">2</cn>
      </apply>
      <cn type="integer">2</cn>
     </apply>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><ci>z</ci><csymbol cd="alg1">one</csymbol></apply>
       <cn type="integer">2</cn>
      </apply>
      <cn type="integer">2</cn>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccosh ( z) , times ( 2 , ln (plus (root (divide (plus ( z, one) , 2 ) , 2 ) , root (divide (minus ( z, one) , 2 ) , 2 ) ) ) ) )

Popcorn

arccosh($z) = 2 * ln(arith1.root(($z + alg1.one) / 2, 2) + arith1.root(($z - alg1.one) / 2, 2))

Rendered Presentation MathML

$$\mathrm{arccosh}\left(z\right)=2\ln \left(\sqrt{\frac{z+1}{2}}+\sqrt{\frac{z-1}{2}}\right)$$

Commented Mathematical property (CMP):

arccosh z = i * (pi - arccos z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccosh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMS cd="nums1" name="pi"/>
      <OMA>
	<OMS cd="transc1" name="arccos"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccosh</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <csymbol cd="nums1">i</csymbol>
   <apply><csymbol cd="arith1">minus</csymbol>
    <csymbol cd="nums1">pi</csymbol>
    <apply><csymbol cd="transc1">arccos</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccosh ( z) , times (i, minus (pi, arccos ( z) ) ) )

Popcorn

arccosh($z) = nums1.i * (nums1.pi - arccos($z))

Rendered Presentation MathML

$$\mathrm{arccosh}\left(z\right)=i(\pi -\arccos \left(z\right))$$

Signatures:

sts

---

[Next: arctanh] [Previous: arcsinh] [Top]

---

arctanh

Role:

application

Description:

This symbol represents the arctanh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arctanh(z) = - i * arctan(i * z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arctanh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="arctan"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arctanh</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">i</csymbol></apply>
   <apply><csymbol cd="transc1">arctan</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arctanh ( z) , times (unary_minus (i) , arctan (times (i, z) ) ) )

Popcorn

arctanh($z) = -(nums1.i) * arctan(nums1.i * $z)

Rendered Presentation MathML

$$\mathrm{arctanh}\left(z\right)=-i\arctan \left(iz\right)$$

Commented Mathematical property (CMP):

for all x where 0 <= x^2 < 1 | arctanh(x) = 1/2 * ln((1 + x)/(1 - x))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="relation1" name="leq"/>
	<OMS cd="alg1" name="zero"/>
	<OMA>
	  <OMS cd="arith1" name="power"/>
	  <OMV name="x"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
	<OMA>
	  <OMS cd="arith1" name="power"/>
	  <OMV name="x"/>
	  <OMI> 2 </OMI>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arctanh"/>
	<OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMA>
          <OMS cd="nums1" name="rational"/>
          <OMI> 1 </OMI>
          <OMI> 2 </OMI>
        </OMA>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="divide"/>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMV name="x"/>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMS cd="alg1" name="one"/>
	      <OMV name="x"/>
	    </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="relation1">leq</csymbol>
     <csymbol cd="alg1">zero</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn type="integer">2</cn></apply>
    </apply>
    <apply><csymbol cd="relation1">lt</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn type="integer">2</cn></apply>
     <csymbol cd="alg1">one</csymbol>
    </apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="transc1">arctanh</csymbol><ci>x</ci></apply>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="nums1">rational</csymbol>
      <cn type="integer">1</cn>
      <cn type="integer">2</cn>
     </apply>
     <apply><csymbol cd="transc1">ln</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><csymbol cd="alg1">one</csymbol></apply>
       <apply><csymbol cd="arith1">minus</csymbol><csymbol cd="alg1">one</csymbol><ci>x</ci></apply>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x ] . (implies (and (leq (zero, power ( x, 2 ) ) , lt (power ( x, 2 ) , one) ) , eq (arctanh ( x) , times (rational ( 1 , 2 ) , ln (divide (plus ( x, one) , minus (one, x) ) ) ) ) ) )

Popcorn

quant1.forall[$x -> alg1.zero <= $x ^ 2 and $x ^ 2 < alg1.one ==> arctanh($x) = 1 // 2 * ln(($x + alg1.one) / (alg1.one - $x))]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x.0\le x^2\wedge x^2<1\Rightarrow \mathrm{arctanh}\left(x\right)=\frac{1}{2}\ln \left(\frac{x+1}{1-x}\right)$$

Signatures:

sts

---

[Next: arcsech] [Previous: arccosh] [Top]

---

arcsech

Role:

application

Description:

This symbol represents the arcsech function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arcsech(z) = 2 ln(\sqrt((1+z)/(2z)) + \sqrt((1-z)/(2z)))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsech"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="plus"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
		<OMI> 2 </OMI>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
		<OMI> 2 </OMI>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arcsech</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <cn type="integer">2</cn>
   <apply><csymbol cd="transc1">ln</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">plus</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
       <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>z</ci></apply>
      </apply>
      <cn type="integer">2</cn>
     </apply>
     <apply><csymbol cd="arith1">root</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
       <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>z</ci></apply>
      </apply>
      <cn type="integer">2</cn>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arcsech ( z) , times ( 2 , ln (plus (root (divide (plus (one, z) , times ( 2 , z) ) , 2 ) , root (divide (minus (one, z) , times ( 2 , z) ) , 2 ) ) ) ) )

Popcorn

arcsech($z) = 2 * ln(arith1.root((alg1.one + $z) / (2 * $z), 2) + arith1.root((alg1.one - $z) / (2 * $z), 2))

Rendered Presentation MathML

$$\mathrm{arcsech}\left(z\right)=2\ln \left(\sqrt{\frac{1+z}{2z}}+\sqrt{\frac{1-z}{2z}}\right)$$

Commented Mathematical property (CMP):

for all x in (0..1] | arcsech x = ln(1/x + (1/(x^2) - 1)^(1/2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS cd="interval1" name="interval_oc"/>
	<OMI> 0 </OMI> <OMI> 1 </OMI>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arcsech"/>
	<OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="power"/>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMA>
	        <OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="x"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMA>
	      <OMS cd="nums1" name="rational"/>
	      <OMI> 1 </OMI> <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="interval1">interval_oc</csymbol>
     <cn type="integer">0</cn>
     <cn type="integer">1</cn>
    </apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="transc1">arcsech</csymbol><ci>x</ci></apply>
    <apply><csymbol cd="transc1">ln</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="alg1">one</csymbol><ci>x</ci></apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol>
        <apply><csymbol cd="arith1">divide</csymbol>
         <csymbol cd="alg1">one</csymbol>
         <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn type="integer">2</cn></apply>
        </apply>
        <csymbol cd="alg1">one</csymbol>
       </apply>
       <apply><csymbol cd="nums1">rational</csymbol>
        <cn type="integer">1</cn>
        <cn type="integer">2</cn>
       </apply>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x ] . (implies (in ( x, interval_oc ( 0 , 1 ) ) , eq (arcsech ( x) , ln (plus (divide (one, x) , power (minus (divide (one, power ( x, 2 ) ) , one) , rational ( 1 , 2 ) ) ) ) ) ) )

Popcorn

quant1.forall[$x -> set1.in($x, interval1.interval_oc(0, 1)) ==> arcsech($x) = ln(alg1.one / $x + (alg1.one / $x ^ 2 - alg1.one) ^ 1 // 2)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x.x\in (0,1]\Rightarrow \mathrm{arcsech}\left(x\right)=\ln \left(\frac{1}{x}+{(\frac{1}{x^2}-1)}^{\frac{1}{2}}\right)$$

Signatures:

sts

---

[Next: arccsch] [Previous: arctanh] [Top]

---

arccsch

Role:

application

Description:

This symbol represents the arccsch function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arccsch(z) = ln(1/z + \sqrt(1+(1/z)^2))

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccsch"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="ln"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="z"/>
	</OMA>
	<OMA>
	  <OMS cd="arith1" name="root"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMA>
	        <OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccsch</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="transc1">ln</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
    <apply><csymbol cd="arith1">root</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol>
      <csymbol cd="alg1">one</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
       <cn type="integer">2</cn>
      </apply>
     </apply>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccsch ( z) , ln (plus (divide (one, z) , root (plus (one, power (divide (one, z) , 2 ) ) , 2 ) ) ) )

Popcorn

arccsch($z) = ln(alg1.one / $z + arith1.root(alg1.one + (alg1.one / $z) ^ 2, 2))

Rendered Presentation MathML

$$\mathrm{arccsch}\left(z\right)=\ln \left(\frac{1}{z}+\sqrt{1+{\left(\frac{1}{z}\right)}^2}\right)$$

Commented Mathematical property (CMP):

arccsch(z) = i * arccsc(i * z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsch"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccsch</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <csymbol cd="nums1">i</csymbol>
   <apply><csymbol cd="transc1">arccsc</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (arccsch ( z) , times (i, arccsc (times (i, z) ) ) )

Popcorn

arccsch($z) = nums1.i * arccsc(nums1.i * $z)

Rendered Presentation MathML

$$\mathrm{arccsch}\left(z\right)=i\mathrm{arccsc}\left(iz\right)$$

Signatures:

sts

---

[Next: arccoth] [Previous: arcsech] [Top]

---

arccoth

Role:

application

Description:

This symbol represents the arccoth function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):

arccoth(z) = (ln(-1-z)-ln(1-z))/2

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccoth"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="arith1" name="minus"/>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="minus"/>
	    <OMA>
	      <OMS cd="arith1" name="unary_minus"/>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMV name="z"/>
          </OMA>
        </OMA>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="minus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="z"/>
          </OMA>
        </OMA>
      </OMA>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="transc1">arccoth</csymbol><ci>z</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="transc1">ln</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol>
      <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="alg1">one</csymbol></apply>
      <ci>z</ci>
     </apply>
    </apply>
    <apply><csymbol cd="transc1">ln</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol><csymbol cd="alg1">one</csymbol><ci>z</ci></apply>
    </apply>
   </apply>
   <cn type="integer">2</cn>
  </apply>
 </apply>
</math>

Prefix

eq (arccoth ( z) , divide (minus (ln (minus (unary_minus (one) , z) ) , ln (minus (one, z) ) ) , 2 ) )

Popcorn

arccoth($z) = (ln( -(alg1.one) - $z) - ln(alg1.one - $z)) / 2

Rendered Presentation MathML

$$\mathrm{arccoth}\left(z\right)=\frac{\ln \left(-1-z\right)-\ln \left(1-z\right)}{2}$$

Commented Mathematical property (CMP):

for all z | if z is not zero then arccoth(z) = i * arccot(i * z)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="z"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="relation1" name="neq"/>
      <OMV name="z"/>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arccoth"/>
	<OMV name="z"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
        <OMA>
          <OMS cd="transc1" name="arccot"/>
	  <OMA>
            <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMV name="z"/>
          </OMA>
        </OMA>
      </OMA> 
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="relation1">neq</csymbol><ci>z</ci><csymbol cd="alg1">zero</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="transc1">arccoth</csymbol><ci>z</ci></apply>
    <apply><csymbol cd="arith1">times</csymbol>
     <csymbol cd="nums1">i</csymbol>
     <apply><csymbol cd="transc1">arccot</csymbol>
      <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>z</ci></apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (neq ( z, zero) , eq (arccoth ( z) , times (i, arccot (times (i, z) ) ) ) ) )

Popcorn

quant1.forall[$z -> $z != alg1.zero ==> arccoth($z) = nums1.i * arccot(nums1.i * $z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\ne 0\Rightarrow \mathrm{arccoth}\left(z\right)=i\mathrm{arccot}\left(iz\right)$$

Signatures:

sts

---

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