OpenMath

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

https://openmath.org

OpenMath Content Dictionary: transc2

Canonical URL:
http://www.openmath.org/cd/transc2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
transc2.ocd
CD as XML Encoded OpenMath:
transc2.omcd
Defines:
arctan, unwind
Date:
2004-03-30
Version:
2 (Revision 1)
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 (James Davenport)
  SourceURL: https://github.com/OpenMath/CDs

This CD holds the definition of a two-argument version of arctan, useful for defining the argument of a complex number, and equivalent to Fortran's ATAN2 function. It also holds a definition of the unwinding number, useful for writing correct relationships between elementary functions.

arctan

Role:
application
Description:

This symbol represents the two-argument arctan function as in Fortran's ATAN2. arctan(x,y) is a value of arctan(y/x). For real x,y arctan(x,y) is positive when y is positive, negative when y is negative. If y is zero, the result is 0 if x is positive, and $\pi$ if x is negative. If x is zero, the result has absolute value $\pi/2$.

Commented Mathematical property (CMP):
x not 0 implies tan(arctan(y,x))=y/x
Formal Mathematical property (FMP):
x ≠ 0 ⇒ tan ⁡ ( arctan ⁡ ( x , y ) ) = y x
Commented Mathematical property (CMP):
$x,y \in {\bf R} \implies -\pi < arctan(y,x)\le\pi$.
Formal Mathematical property (FMP):
x ∈ R ∧ y ∈ R ⇒ arctan ⁡ ( x , y ) ∈ ( - π , π ]
Commented Mathematical property (CMP):
$Re(y)>0 \implies Re(arctan(y,x))>0$.
Formal Mathematical property (FMP):
real ⁡ ( y ) > 0 ⇒ real ⁡ ( arctan ⁡ ( x , y ) ) > 0
Commented Mathematical property (CMP):
$Re(y) < 0 \implies Re(arctan(y,x)) < 0$.
Formal Mathematical property (FMP):
real ⁡ ( y ) < 0 ⇒ real ⁡ ( arctan ⁡ ( x , y ) ) < 0
Commented Mathematical property (CMP):
$Re(y)=0 and Re(x)>0 \implies Re(arctan(y,x))=0$.
Formal Mathematical property (FMP):
real ⁡ ( y ) = 0 ∧ real ⁡ ( x ) > 0 ⇒ real ⁡ ( arctan ⁡ ( x , y ) ) < 0
Commented Mathematical property (CMP):
$Re(y)=0 and Re(x) < 0 \implies Re(arctan(y,x))=\pi$.
Formal Mathematical property (FMP):
real ⁡ ( y ) = 0 ∧ real ⁡ ( x ) < 0 ⇒ real ⁡ ( arctan ⁡ ( x , y ) ) < π
Commented Mathematical property (CMP):
$x=0 \implies |arctan(y,x)|=\pi$.
Formal Mathematical property (FMP):
x ⇒ | arctan ⁡ ( x , y ) | = π 2
Signatures:
sts

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unwind

Role:
application
Description:

The unwinding number denotes the extent to which $z=\ln\exp z$ is not true. It was orignally defined in Corless,R.M. & Jeffrey,D.J., The Unwinding Number. SIGSAM Bulletin 30(1996) 2, pp. 28-35. However, we take the definition (which has a change of sign) from Corless,R.M., Davenport,J.H., Jeffrey,D.J. & Watt,S.M., According to Abramowitz and Stegun. SIGSAM Bulletin 34(2000) 2, pp. 58--65. Note that the symbol is normally denoted by ${\cal K}$.

Commented Mathematical property (CMP):
unwind(z)=(z-ln exp z)/(2pi i)
Formal Mathematical property (FMP):
unwind ⁡ ( z ) = z - ln ⁡ ( exp ⁡ ( z ) ) 2 ⁢ π ⁢ i
Commented Mathematical property (CMP):
unwind(z)=ceiling((Im z - pi)/(2pi))
Formal Mathematical property (FMP):
unwind ⁡ ( z ) = ceiling ⁡ ( imaginary ⁡ ( z ) - π 2 ⁢ π )
Commented Mathematical property (CMP):
z in C implies unwind(z) in Z
Formal Mathematical property (FMP):
z ∈ C ⇒ unwind ⁡ ( z ) ∈ Z 
    \arcsin z = \arctan\frac z{\sqrt{1-z^2}} +\pi\K(-\ln(1+z))-\pi\K(-\ln(1-z)).
Example:
arcsin ⁡ ( z ) = arctan ⁡ ( z 1 - z 2 ) + π ⁢ ( unwind ⁡ ( - ln ⁡ ( 1 + z ) ) - unwind ⁡ ( - ln ⁡ ( 1 - z ) ) )
Signatures:
sts

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