Home Overview Documents Content Dictionaries Software & Tools The OpenMath Society OpenMath Projects OpenMath Discussion Lists OpenMath Meetings Links

# 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
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 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):
$\forall x,y.x+yi=x+iy$
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):
$\forall x,y.x=\mathrm{real}\left(x+yi\right)$
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):
$\forall x,y.y=\mathrm{imaginary}\left(x+yi\right)$
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):
$\forall r,a.r{e}^{ai}=r\mathrm{exp}\left(ai\right)$
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):
$\forall x,y,r,a.r\mathrm{sin}\left(a\right)=y\wedge r\mathrm{cos}\left(a\right)=x⇒r{e}^{ai}=x+yi$
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):
$\forall x.a\in \mathbb{R}\wedge k\in \mathbb{Z}⇒x{e}^{ai}=x{e}^{\left(a+2\pi k\right)i}$
Example:
i = complex_polar(1,pi/2)
$i={e}^{\frac{\pi }{2}i}$
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):
$\forall r,a.\mathrm{argument}\left(r{e}^{ai}\right)=a$
Commented Mathematical property (CMP):
the argument of x+i*y = arctan(y/x) (if x is positive)
Formal Mathematical property (FMP):
$x>0⇒\mathrm{argument}\left(x+yi\right)=\mathrm{arctan}\left(\frac{y}{x}\right)$
Commented Mathematical property (CMP):
the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2)
Formal Mathematical property (FMP):
$\mathrm{argument}\left(x+yi\right)=\mathrm{arctan}\left(y,x\right)$
Signatures:
sts

 [Next: conjugate] [Previous: complex_polar] [Top]

## 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):
$a\in \mathbb{C}⇒\left(\overline{a}+a\right)\in \mathbb{R}$
Signatures:
sts

 [First: complex_cartesian] [Previous: argument] [Top]

 Home Overview Documents Content Dictionaries Software & Tools The OpenMath Society OpenMath Projects OpenMath Discussion Lists OpenMath Meetings Links