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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 1)
Review Date:
2006-03-30
Status:
experimental 

     This document is distributed in the hope that it will be useful, 
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     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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          both this document and the derived work are Content Dictionaries
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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):