OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

https://openmath.org

OpenMath Content Dictionary: transc3

Canonical URL:
http://www.openmath.org/cd/transc3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
transc3.ocd
CD as XML Encoded OpenMath:
transc3.omcd
Defines:
arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, ln, log
Date:
2004-03-30
Version:
2 (Revision 2)
Review Date:
2017-12-31
Status:
experimental

     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 and related functions. They are defined as multi-valued functions with precise reductions to logs in the case of inverse functions. Note that we use the same names as in the single-valued case, even though it would be traditional to render them with capital letters. In sum <OMS cd="transc3" name="ln"/> is multi-valued, while <OMS cd="transc1" name="ln"/> is single-valued. Note that in many cases A+S only states the log restrictions under some circumstances: JHD has proved (22.8.2002) all the inverse trig. ones

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 is equivalent to b in Log_a c
Formal Mathematical property (FMP):
log ⁡ ( a , c ) = { b ∈ C | a b = c }
Example:
log 100 to base 10 (which is {2+2n\pi i}).
log ⁡ ( 10 , 100 )
Signatures:
sts

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

ln

Role:
application
Description:

This symbol represents the ln function (natural logarithm) as a multivalued function.

Commented Mathematical property (CMP):
y in Ln(x) <=> exp(y)=x
Formal Mathematical property (FMP):
ln ⁡ ( x ) = { y ∈ C | exp ⁡ ( y ) = x }
Formal Mathematical property (FMP):
ln ⁡ ( x ) = { ln ⁡ ( x ) + 2 ⁢ n ⁢ π ⁢ i | n ∈ Z }
Example:
Ln 1 (which is 0+2n\pi i).
ln ⁡ ( 1 )
Signatures:
sts

[Next: arcsin] [Previous: log] [Top]

arcsin

Role:
application
Description:

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

Commented Mathematical property (CMP):
y in Arcsin(x) <=> sin(y)=x
Formal Mathematical property (FMP):
arcsin ⁡ ( x ) = { y ∈ C | sin ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arcsin(z) = -i ln (sqrt(1-z^2)+iz), but the multivalued equivalent is Arcsin(z) = -i Ln (Sqrt(1-z^2)+iz), which we translate into OpenMath as Arcsin(z) = -i [ Ln (sqrt(1-z^2)+iz) union Ln (-sqrt(1-z^2)+iz)], 
Only stated in A+S for \z^2|\le 1, but proved for all z in JHD's OpenMath
deliverable.
Formal Mathematical property (FMP):
arcsin ⁡ ( z ) = { - i ⁢ y | y ∈ ln ⁡ ( 1 - z 2 + i ⁢ z ) ∪ ln ⁡ ( - 1 - z 2 + i ⁢ z ) }
Signatures:
sts

[Next: arccos] [Previous: ln] [Top]

arccos

Role:
application
Description:

This symbol represents the arccos function. This is the multivalued inverse of the cos function.

Commented Mathematical property (CMP):
y in Arccos(x) <=> cos(y)=x
Formal Mathematical property (FMP):
arccos ⁡ ( x ) = { y ∈ C | cos ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arccos(z) = -i ln(z+i \sqrt(1-z^2)), so the multi-valued equivalent is Arccos(z) = -i Ln(z+i \Sqrt(1-z^2)), encoded as Arccos(z) = -i(ln(z+i \sqrt(1-z^2)) union ln(z-i \sqrt(1-z^2))) 
Only stated in A+S for \z^2|\le 1, but proved for all z in JHD's OpenMath
deliverable.
Formal Mathematical property (FMP):
arccos ⁡ ( z ) = { - i ⁢ y | y ∈ ln ⁡ ( i ⁢ 1 - z 2 + z ) ∪ ln ⁡ ( i ⁢ 1 - z 2 - z ) }
Signatures:
sts

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

arctan

Role:
application
Description:

This symbol represents the arctan function. This is the multi-valued inverse of the tan function.

Commented Mathematical property (CMP):
y in Arctan(x) <=> tan(y)=x
Formal Mathematical property (FMP):
arctan ⁡ ( x ) = { y ∈ C | tan ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arctan(z) = (i/2)ln((1-iz)/(1+iz)), so the multi-valued equivalent is Arctan(z) = (i/2)Ln((1-iz)/(1+iz)),
Formal Mathematical property (FMP):
arctan ⁡ ( z ) = { y ⁢ i 2 | y ∈ ln ⁡ ( 1 - i ⁢ z 1 + i ⁢ z ) }
Signatures:
sts

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

arcsec

Role:
application
Description:

This symbol represents the multivalued arcsec function as the inverse of sec.

Commented Mathematical property (CMP):
y in Arcsec(x) <=> sec(y)=x
Formal Mathematical property (FMP):
arcsec ⁡ ( x ) = { y ∈ C | sec ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arcsec(z) = -i ln(1/z+i \sqrt(1-1/z^2)), so the multi-valued equivalent is Arcsec(z) = -i Ln(1/z+i \Sqrt(1-1/z^2)), encoded as Arcsec(z) = -i(ln(1/z+i \sqrt(1-1/z^2)) union ln(1/z-i \sqrt(1-1/z^2)))
Formal Mathematical property (FMP):
arcsec ⁡ ( z ) = { - i ⁢ y | y ∈ ln ⁡ ( i ⁢ 1 - 1 z 2 + 1 z ) ∪ ln ⁡ ( i ⁢ 1 - 1 z 2 - 1 z ) }
Signatures:
sts

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

arccsc

Role:
application
Description:

This symbol represents the multivalued arccsc function as the inverse of csc.

Commented Mathematical property (CMP):
y in Arccsc(x) <=> csc(y)=x
Formal Mathematical property (FMP):
arccsc ⁡ ( x ) = { y ∈ C | csc ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arccsc(z) = -i ln (sqrt(1-1/z^2)+i/z), but the multivalued equivalent is Arccsc(z) = -i Ln (Sqrt(1-1/z^2)+i/z), which we translate into OpenMath as Arccsc(z) = -i [ Ln (sqrt(1-1/z^2)+i/z) union Ln (-sqrt(1-1/z^2)+i/z),
Formal Mathematical property (FMP):
arccsc ⁡ ( z ) = { - i ⁢ y | y ∈ ln ⁡ ( 1 - 1 z 2 + i ⁢ 1 z ) ∪ ln ⁡ ( - 1 - 1 z 2 + i ⁢ 1 z ) }
Signatures:
sts

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

arccot

Role:
application
Description:

This symbol represents the multi-valued arccot function as the inverse of cot

Commented Mathematical property (CMP):
y in Arccot(x) <=> cot(y)=x
Formal Mathematical property (FMP):
arccot ⁡ ( x ) = { y ∈ C | cot ⁡ ( y ) = x }
Commented Mathematical property (CMP):
arccot(-z) = - arccot(z)
Formal Mathematical property (FMP):
arccot ⁡ ( - z ) = { - x | x ∈ arccot ⁡ ( z ) }
Commented Mathematical property (CMP):
arccot(z) = (i/2)ln((1+iz)/(1-iz)), so the multi-valued equivalent is Arccot(z) = (i/2)Ln((1+iz)/(1-iz)),
Formal Mathematical property (FMP):
arccot ⁡ ( z ) = { y ⁢ i 2 | y ∈ ln ⁡ ( 1 + i ⁢ z 1 - i ⁢ z ) }
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):
y in Arcsinh(x) <=> sinh(y)=x
Formal Mathematical property (FMP):
arcsinh ⁡ ( x ) = { y ∈ C | sinh ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arcsinh z = ln(z +-\sqrt(1+z^2))
Formal Mathematical property (FMP):
arcsinh ⁡ ( z ) = ln ⁡ ( z + 1 + z 2 ) ∪ ln ⁡ ( z - 1 + z 2 )
Commented Mathematical property (CMP):
Arcsinh(z) = - i * Arcsin(i * z)
Formal Mathematical property (FMP):
arcsinh ⁡ ( z ) = { - i ⁢ y | y ∈ arcsin ⁡ ( i ⁢ z ) }
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):
y in Arccosh(x) <=> cosh(y)=x
Formal Mathematical property (FMP):
arccosh ⁡ ( x ) = { y ∈ C | cosh ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arccosh z = ln(z +-\sqrt(z^2-1))
Formal Mathematical property (FMP):
arccosh ⁡ ( z ) = ln ⁡ ( z + z 2 - 1 ) ∪ ln ⁡ ( z - z 2 - 1 )
Commented Mathematical property (CMP):
Arccosh(z) = i * Arccos(i * z) 
 
  A+S says +/- i ..., but this is irrelevant since Arccos(iz)=-Arccos(iz)
Formal Mathematical property (FMP):
arccosh ⁡ ( z ) = { i ⁢ y | y ∈ arccos ⁡ ( i ⁢ z ) }
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):
y in Arctanh(x) <=> tanh(y)=x
Formal Mathematical property (FMP):
arctanh ⁡ ( x ) = { y ∈ C | tanh ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arctanh(z) = - i * Arctan(i * z)
Formal Mathematical property (FMP):
arctanh ⁡ ( z ) = { - i ⁢ y | y ∈ arctan ⁡ ( i ⁢ z ) }
Commented Mathematical property (CMP):
for all x arctanh(x) = 1/2 * ln((1 + x)/(1 - x)) 
  The condition 0\le x^2 < 1 in A+S is not necessary
  The proof for Arctan is in JHD's OpenMath deliverable
Formal Mathematical property (FMP):
arctanh ⁡ ( z ) = { 1 2 ⁢ y | y ∈ ln ⁡ ( z + 1 1 - z ) }
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):
y in Arcsech(x) <=> sech(y)=x
Formal Mathematical property (FMP):
arcsech ⁡ ( x ) = { y ∈ C | sech ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arcsech z = ln(1/z +-\sqrt(1/z^2-1))
Formal Mathematical property (FMP):
arcsech ⁡ ( z ) = ln ⁡ ( 1 z + 1 z 2 - 1 ) ∪ ln ⁡ ( 1 z - 1 z 2 - 1 )
Commented Mathematical property (CMP):
Arcsech(z) = i * Arcsec(i * z) 
 
  A+S says +/- i ..., but this is irrelevant since Arcsec(iz)=-Arcsec(iz)
Formal Mathematical property (FMP):
arcsech ⁡ ( z ) = { i ⁢ y | y ∈ arcsec ⁡ ( i ⁢ z ) }
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):
y in Arccsch(x) <=> csch(y)=x
Formal Mathematical property (FMP):
arccsch ⁡ ( x ) = { y ∈ C | csch ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arccsch z = ln(1/z +-\sqrt(1+1/z^2))
Formal Mathematical property (FMP):
arccsch ⁡ ( z ) = ln ⁡ ( 1 z + 1 + 1 z 2 ) ∪ ln ⁡ ( 1 z - 1 + 1 z 2 )
Commented Mathematical property (CMP):
Arccsch(z) = i * Arccsc(i * z)
Formal Mathematical property (FMP):
arcsinh ⁡ ( z ) = { i ⁢ y | y ∈ arcsin ⁡ ( i ⁢ z ) }
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):
y in Arccoth(x) <=> coth(y)=x
Formal Mathematical property (FMP):
arccoth ⁡ ( x ) = { y ∈ C | coth ⁡ ( y ) = x }
Commented Mathematical property (CMP):
Arccoth(z) = i * Arccot(i * z)
Formal Mathematical property (FMP):
arccoth ⁡ ( z ) = { - i ⁢ y | y ∈ arccot ⁡ ( i ⁢ z ) }
Commented Mathematical property (CMP):
for all x arccoth(x) = 1/2 * ln((x + 1)/(x - 1)) 
  The condition 0\le x^2 < 1 in A+S is not necessary
  The proof for Arctan is in JHD's OpenMath deliverable
Formal Mathematical property (FMP):
arccoth ⁡ ( z ) = { 1 2 ⁢ y | y ∈ ln ⁡ ( z + 1 z - 1 ) }
Signatures:
sts

[First: log] [Previous: arccsch] [Top]