OpenMath Content Dictionary: complex1

Canonical URL:

http://www.openmath.org/cd/complex1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

complex1.ocd

CD as XML Encoded OpenMath:

complex1.omcd

Defines:

argument, complex_cartesian, complex_polar, conjugate, imaginary, real

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 is intended to be `compatible' with the MathML view of operations on and constructors for complex numbers.

---

complex_cartesian

Role:

application

Description:

This symbol represents a constructor function for complex numbers specified as the Cartesian coordinates of the relevant point on the complex plane. It takes two arguments, the first is a number x to denote the real part and the second a number y to denote the imaginary part of the complex number x + i y. (Where i is the square root of -1.)

Commented Mathematical property (CMP):

for all x,y | complex_cartesian(x,y) = x + iy

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"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="complex1" name="complex_cartesian"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
        <OMA>
          <OMS cd="arith1" name="plus"/>
          <OMV name="x"/>
          <OMA>
            <OMS cd="arith1" name="times"/>
            <OMS cd="nums1" name="i"/>
            <OMV name="y"/>
          </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>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   <apply><csymbol cd="arith1">plus</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x y ] . (eq (complex_cartesian ( x, y) , plus ( x, times (i, y) ) ) )

Popcorn

quant1.forall[$x, $y -> $x | $y = $x + nums1.i * $y]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x,y.x+yi=x+iy$$

Signatures:

sts

---

[Next: real] [Last: conjugate] [Top]

---

real

Role:

application

Description:

This represents the real part of a complex number

Commented Mathematical property (CMP):

for all x,y | x = real(x+iy)

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"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="x"/>
        <OMA>
          <OMS name="real" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </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>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="complex1">real</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x y ] . (eq ( x, real (complex_cartesian ( x, y) ) ) )

Popcorn

quant1.forall[$x, $y -> $x = complex1.real($x | $y)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x,y.x=\mathrm{real}\left(x+yi\right)$$

Signatures:

sts

---

[Next: imaginary] [Previous: complex_cartesian] [Top]

---

imaginary

Role:

application

Description:

This represents the imaginary part of a complex number

Commented Mathematical property (CMP):

for all x,y | y = imaginary(x+iy)

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"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="y"/>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </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>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>y</ci>
   <apply><csymbol cd="complex1">imaginary</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x y ] . (eq ( y, imaginary (complex_cartesian ( x, y) ) ) )

Popcorn

quant1.forall[$x, $y -> $y = complex1.imaginary($x | $y)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x,y.y=\mathrm{imaginary}\left(x+yi\right)$$

Signatures:

sts

---

[Next: complex_polar] [Previous: real] [Top]

---

complex_polar

Role:

application

Description:

This symbol represents a constructor function for complex numbers specified as the polar coordinates of the relevant point on the complex plane. It takes two arguments, the first is a nonnegative number r to denote the magnitude and the second a number theta (given in radians) to denote the argument of the complex number r e^(i theta). (i and e are defined as in this CD).

Commented Mathematical property (CMP):

for all r,a | complex_polar(r,a) = r*e^(a*i)

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="r"/>
      <OMV name="a"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMV name="r"/>
	<OMA>
	  <OMS cd="transc1" name="exp"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMV name="a"/>
	    <OMS cd="nums1" name="i"/>
	  </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>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <ci>r</ci>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><ci>a</ci><csymbol cd="nums1">i</csymbol></apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ r a ] . (eq (complex_polar ( r, a) , times ( r, exp (times ( a, i) ) ) ) )

Popcorn

quant1.forall[$r, $a -> complex1.complex_polar($r, $a) = $r * exp($a * nums1.i)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}r,a.r{e}^{ai}=r\exp \left(ai\right)$$

Commented Mathematical property (CMP):

for all x,y,r,a | (r sin a = y and r cos a = x) implies (complex_polar(r,a) = complex_cartesian(x,y)

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"/>
    <OMV name="y"/>
    <OMV name="r"/>
    <OMV name="a"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="times"/>
	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="y"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="times"/>
	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </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>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <ci>r</ci>
      <apply><csymbol cd="transc1">sin</csymbol><ci>a</ci></apply>
     </apply>
     <ci>y</ci>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <ci>r</ci>
      <apply><csymbol cd="transc1">cos</csymbol><ci>a</ci></apply>
     </apply>
     <ci>x</ci>
    </apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x y r a ] . (implies (and (eq (times ( r, sin ( a) ) , y) , eq (times ( r, cos ( a) ) , x) ) , eq (complex_polar ( r, a) , complex_cartesian ( x, y) ) ) )

Popcorn

quant1.forall[$x, $y, $r, $a -> $r * sin($a) = $y and $r * cos($a) = $x ==> complex1.complex_polar($r, $a) = $x | $y]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x,y,r,a.r\sin \left(a\right)=y\wedge r\cos \left(a\right)=x\Rightarrow r{e}^{ai}=x+yi$$

Commented Mathematical property (CMP):

for all x | if a is a real number and k is an integer then complex_polar(x,a) = complex_polar(x,a+2*pi*k)

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="set1" name="in"/>
	<OMV name="a"/>
	<OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="k"/>
	<OMS cd="setname1" name="Z"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="a"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMI> 2 </OMI>
	    <OMS cd="nums1" name="pi"/>
	    <OMV name="k"/>
	  </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="set1">in</csymbol><ci>a</ci><csymbol cd="setname1">R</csymbol></apply>
    <apply><csymbol cd="set1">in</csymbol><ci>k</ci><csymbol cd="setname1">Z</csymbol></apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>x</ci><ci>a</ci></apply>
    <apply><csymbol cd="complex1">complex_polar</csymbol>
     <ci>x</ci>
     <apply><csymbol cd="arith1">plus</csymbol>
      <ci>a</ci>
      <apply><csymbol cd="arith1">times</csymbol>
       <cn type="integer">2</cn>
       <csymbol cd="nums1">pi</csymbol>
       <ci>k</ci>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ x ] . (implies (and (in ( a, R) , in ( k, Z) ) , eq (complex_polar ( x, a) , complex_polar ( x, plus ( a, times ( 2 , pi, k) ) ) ) ) )

Popcorn

quant1.forall[$x -> set1.in($a, setname1.R) and set1.in($k, setname1.Z) ==> complex1.complex_polar($x, $a) = complex1.complex_polar($x, $a + 2 * nums1.pi * $k)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}x.a\in \mathbb{R}\wedge k\in \mathbb{Z}\Rightarrow x{e}^{ai}=x{e}^{(a+2\pi k)i}$$

Example:

i = complex_polar(1,pi/2)

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"/>
  <OMS cd="nums1" name="i"/>
  <OMA>
    <OMS cd="complex1" name="complex_polar"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="nums1" name="pi"/>
      <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>
  <csymbol cd="nums1">i</csymbol>
  <apply><csymbol cd="complex1">complex_polar</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <csymbol cd="nums1">pi</csymbol>
    <cn type="integer">2</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (i, complex_polar (one, divide (pi, 2 ) ) )

Popcorn

nums1.i = complex1.complex_polar(alg1.one, nums1.pi / 2)

Rendered Presentation MathML

$$i=e^{\frac{\pi }{2}i}$$

Signatures:

sts

---

[Next: argument] [Previous: imaginary] [Top]

---

argument

Role:

application

Description:

This symbol represents the unary function which returns the argument of a complex number, viz. the angle which a straight line drawn from the number to zero makes with the Real line (measured anti-clockwise). The argument to the symbol is the complex number whos argument is being taken.

Commented Mathematical property (CMP):

for all r,a | argument(complex_polar(r,a)=a)

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="r"/>
      <OMV name="a"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="argument"/>
        <OMA>
          <OMS cd="complex1" name="complex_polar"/>
          <OMV name="r"/>
          <OMV name="a"/>
        </OMA>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">argument</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
   </apply>
   <ci>a</ci>
  </apply>
 </bind>
</math>

Prefix

forall [ r a ] . (eq (argument (complex_polar ( r, a) ) , a) )

Popcorn

quant1.forall[$r, $a -> complex1.argument(complex1.complex_polar($r, $a)) = $a]

Rendered Presentation MathML

$$\forall \hspace{0.17em}r,a.\mathrm{argument}\left(r{e}^{ai}\right)=a$$

Commented Mathematical property (CMP):

the argument of x+i*y = arctan(y/x) (if x is positive)

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="gt"/>
      <OMV name="x"/>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="argument"/>
        <OMA>
          <OMS cd="complex1" name="complex_cartesian"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="arctan"/>
        <OMA>
          <OMS cd="arith1" name="divide"/>
	  <OMV name="y"/>
	  <OMV name="x"/>
        </OMA>
      </OMA>
    </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">gt</csymbol><ci>x</ci><csymbol cd="alg1">zero</csymbol></apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">argument</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
   <apply><csymbol cd="transc1">arctan</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><ci>y</ci><ci>x</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (gt ( x, zero) , eq (argument (complex_cartesian ( x, y) ) , arctan (divide ( y, x) ) ) )

Popcorn

$x > alg1.zero ==> complex1.argument($x | $y) = arctan($y / $x)

Rendered Presentation MathML

$$x>0\Rightarrow \mathrm{argument}\left(x+yi\right)=\arctan \left(\frac{y}{x}\right)$$

Commented Mathematical property (CMP):

the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2)

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="complex1" name="argument"/>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="transc2" name="arctan"/>
      <OMV name="y"/>
      <OMV name="x"/>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="complex1">argument</csymbol>
   <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
  </apply>
  <apply><csymbol cd="transc2">arctan</csymbol><ci>y</ci><ci>x</ci></apply>
 </apply>
</math>

Prefix

eq (argument (complex_cartesian ( x, y) ) , arctan ( y, x) )

Popcorn

complex1.argument($x | $y) = transc2.arctan($y, $x)

Rendered Presentation MathML

$$\mathrm{argument}\left(x+yi\right)=\arctan (y,x)$$

Signatures:

sts

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conjugate

Role:

application

Description:

A unary operator representing the complex conjugate of its argument.

Commented Mathematical property (CMP):

if a is a complex number then (conjugate(a) + a) is a real number

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="a"/>
      <OMS cd="setname1" name="C"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMA>
          <OMS cd="complex1" name="conjugate"/>
          <OMV name="a"/>
        </OMA>
        <OMV name="a"/>
      </OMA>
      <OMS cd="setname1" name="R"/>
    </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>a</ci><csymbol cd="setname1">C</csymbol></apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="complex1">conjugate</csymbol><ci>a</ci></apply>
    <ci>a</ci>
   </apply>
   <csymbol cd="setname1">R</csymbol>
  </apply>
 </apply>
</math>

Prefix

implies (in ( a, C) , in (plus (conjugate ( a) , a) , R) )

Popcorn

set1.in($a, setname1.C) ==> set1.in(complex1.conjugate($a) + $a, setname1.R)

Rendered Presentation MathML

$$a\in \mathbb{C}\Rightarrow (\overline{a}+a)\in \mathbb{R}$$

Signatures:

sts

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