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# 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
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.
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.
society at http://www.openmath.org.


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):
${a}^{b}=c⇒{\mathrm{log}}_{a}\left(c\right)=b$
Example:
log 100 to base 10 (which is 2).
${\mathrm{log}}_{10}\left(100\right)$
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):
$-\pi <\mathrm{imaginary}\left(\mathrm{ln}\left(x\right)\right)\wedge \mathrm{imaginary}\left(\mathrm{ln}\left(x\right)\right)\le \pi$
Example:
ln 1 (which is 0).
$\mathrm{ln}\left(1\right)$
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):
$\forall k.k\in \mathbb{Z}⇒\mathrm{exp}\left(z+2\pi ki\right)=\mathrm{exp}\left(z\right)$
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):
$\mathrm{sin}\left(x\right)=\frac{\mathrm{exp}\left(ix\right)-\mathrm{exp}\left(-ix\right)}{2i}$
Commented Mathematical property (CMP):
sin(A + B) = sin A cos B + cos A sin B
Formal Mathematical property (FMP):
$\mathrm{sin}\left(A+B\right)=\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)$
Commented Mathematical property (CMP):
sin A = - sin(-A)
Formal Mathematical property (FMP):
$\mathrm{sin}\left(A\right)=-\mathrm{sin}\left(-A\right)$
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):
$\mathrm{cos}\left(x\right)=\frac{\mathrm{exp}\left(ix\right)+\mathrm{exp}\left(-ix\right)}{2}$
Commented Mathematical property (CMP):
cos 2A = cos^2 A - sin^2 A
Formal Mathematical property (FMP):
$\mathrm{cos}\left(2A\right)={\mathrm{cos}\left(A\right)}^{2}-{\mathrm{sin}\left(A\right)}^{2}$
Commented Mathematical property (CMP):
cos A = cos(-A)
Formal Mathematical property (FMP):
$\mathrm{cos}\left(A\right)=\mathrm{cos}\left(-A\right)$
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):
$\mathrm{tan}\left(A\right)=\frac{\mathrm{sin}\left(A\right)}{\mathrm{cos}\left(A\right)}$
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):
$\mathrm{sec}\left(A\right)=\frac{1}{\mathrm{cos}\left(A\right)}$
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):
$\mathrm{csc}\left(A\right)=\frac{1}{\mathrm{sin}\left(A\right)}$
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):
$\mathrm{cot}\left(A\right)=\frac{1}{\mathrm{tan}\left(A\right)}$
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):
$\mathrm{sinh}\left(A\right)=\frac{1}{2}\left({e}^{A}-{e}^{-A}\right)$
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):
$\mathrm{cosh}\left(A\right)=\frac{1}{2}\left({e}^{A}+{e}^{-A}\right)$
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):
$\mathrm{tanh}\left(A\right)=\frac{\mathrm{sinh}\left(A\right)}{\mathrm{cosh}\left(A\right)}$
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):
$\mathrm{sech}\left(A\right)=\frac{1}{\mathrm{cosh}\left(A\right)}$
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):
$\mathrm{csch}\left(A\right)=\frac{1}{\mathrm{sinh}\left(A\right)}$
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):
$\mathrm{coth}\left(A\right)=\frac{1}{\mathrm{tanh}\left(A\right)}$
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):
$\mathrm{arcsin}\left(z\right)=-i\mathrm{ln}\left(\sqrt{1-{z}^{2}}+iz\right)$
Commented Mathematical property (CMP):
x in [-(pi/2),(pi/2)] implies arcsin(sin x) = x
Formal Mathematical property (FMP):
$x\in \left[-\left(\frac{\pi }{2}\right),\frac{\pi }{2}\right]⇒\mathrm{arcsin}\left(\mathrm{sin}\left(x\right)\right)=x$
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):
$\mathrm{arccos}\left(z\right)=-i\mathrm{ln}\left(z+i\sqrt{1-{z}^{2}}\right)$
Commented Mathematical property (CMP):
x in [0,pi] implies arccos(cos x) = x
Formal Mathematical property (FMP):
$x\in \left[0,\pi \right]⇒\mathrm{arccos}\left(\mathrm{cos}\left(x\right)\right)=x$
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):
$\mathrm{arctan}\left(z\right)=\frac{i}{2}\mathrm{ln}\left(\frac{1-iz}{1+iz}\right)$
Commented Mathematical property (CMP):
x in (-(pi/2),(pi/2)) implies arctan(tan x) = x
Formal Mathematical property (FMP):
$x\in \left(-\left(\frac{\pi }{2}\right),\frac{\pi }{2}\right)⇒\mathrm{arctan}\left(\mathrm{tan}\left(x\right)\right)=x$
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):
$\mathrm{arcsec}\left(z\right)=-i\mathrm{ln}\left(\frac{1}{z}+i\sqrt{1-\frac{1}{{z}^{2}}}\right)$
Commented Mathematical property (CMP):
for all z | arcsec z = i * arcsech z
Formal Mathematical property (FMP):
$\forall z.\mathrm{arcsec}\left(z\right)=i\mathrm{arcsech}\left(z\right)$
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):
$\mathrm{arccsc}\left(z\right)=-i\mathrm{ln}\left(\frac{i}{z}+\sqrt{1-\frac{1}{{z}^{2}}}\right)$
Commented Mathematical property (CMP):
arccsc(z) = i * arccsch(i * z)
Formal Mathematical property (FMP):
$\mathrm{arccsc}\left(z\right)=i\mathrm{arccsch}\left(iz\right)$
Commented Mathematical property (CMP):
arccsc(-z) = - arccsc(z)
Formal Mathematical property (FMP):
$\mathrm{arccsc}\left(-z\right)=-\mathrm{arccsc}\left(z\right)$
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):
$\mathrm{arccot}\left(-z\right)=-\mathrm{arccot}\left(z\right)$
Commented Mathematical property (CMP):
arccot(x) = (i/2) * ln ((x - i)/(x + i))
Formal Mathematical property (FMP):
$\mathrm{arccot}\left(x\right)=\frac{i}{2}\mathrm{ln}\left(\frac{x-i}{x+i}\right)$
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):
$\mathrm{arcsinh}\left(z\right)=\mathrm{ln}\left(z+\sqrt{1+{z}^{2}}\right)$
Commented Mathematical property (CMP):
arcsinh(z) = - i * arcsin(i * z)
Formal Mathematical property (FMP):
$\mathrm{arcsinh}\left(z\right)=-i\mathrm{arcsin}\left(iz\right)$
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):
$\mathrm{arccosh}\left(z\right)=2\mathrm{ln}\left(\sqrt{\frac{z+1}{2}}+\sqrt{\frac{z-1}{2}}\right)$
Commented Mathematical property (CMP):
arccosh z = i * (pi - arccos z)
Formal Mathematical property (FMP):
$\mathrm{arccosh}\left(z\right)=i\left(\pi -\mathrm{arccos}\left(z\right)\right)$
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):
$\mathrm{arctanh}\left(z\right)=-i\mathrm{arctan}\left(iz\right)$
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):
$\forall x.0\le {x}^{2}\wedge {x}^{2}<1⇒\mathrm{arctanh}\left(x\right)=\frac{1}{2}\mathrm{ln}\left(\frac{x+1}{1-x}\right)$
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):
$\mathrm{arcsech}\left(z\right)=2\mathrm{ln}\left(\sqrt{\frac{1+z}{2z}}+\sqrt{\frac{1-z}{2z}}\right)$
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):
$\forall x.x\in \left(0,1\right]⇒\mathrm{arcsech}\left(x\right)=\mathrm{ln}\left(\frac{1}{x}+{\left(\frac{1}{{x}^{2}}-1\right)}^{\frac{1}{2}}\right)$
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):
$\mathrm{arccsch}\left(z\right)=\mathrm{ln}\left(\frac{1}{z}+\sqrt{1+{\left(\frac{1}{z}\right)}^{2}}\right)$
Commented Mathematical property (CMP):
arccsch(z) = i * arccsc(i * z)
Formal Mathematical property (FMP):
$\mathrm{arccsch}\left(z\right)=i\mathrm{arccsc}\left(iz\right)$
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):
$\mathrm{arccoth}\left(z\right)=\frac{\mathrm{ln}\left(-1-z\right)-\mathrm{ln}\left(1-z\right)}{2}$
Commented Mathematical property (CMP):
for all z | if z is not zero then arccoth(z) = i * arccot(i * z)
Formal Mathematical property (FMP):
$\forall z.z\ne 0⇒\mathrm{arccoth}\left(z\right)=i\mathrm{arccot}\left(iz\right)$
Signatures:
sts

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

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