complex1.ocd

<CD xmlns="http://www.openmath.org/OpenMathCD">

<CDComment>

     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.
</CDComment>

<CDName>complex1</CDName>
<CDBase>http://www.openmath.org/cd</CDBase>
<CDURL>http://www.openmath.org/cd/complex1.ocd</CDURL>
<CDReviewDate>2006-03-30</CDReviewDate>
<CDDate>2004-03-30</CDDate>
<CDVersion>3</CDVersion>
<CDRevision>1</CDRevision>
<CDComment>
  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
</CDComment>
<CDStatus>official</CDStatus>
<Description>
    This CD is intended to be `compatible' with the MathML view of
    operations on and constructors for  complex numbers.
</Description>

<CDDefinition>
<Name>complex_cartesian</Name>
<Role>application</Role>
<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.)
</Description>
<CMP>
for all x,y | complex_cartesian(x,y) = x + iy
</CMP>
<FMP>
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      <OMA>
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        <OMA>
          <OMS cd="complex1" name="complex_cartesian"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
        <OMA>
          <OMS cd="arith1" name="plus"/>
          <OMV name="x"/>
          <OMA>
            <OMS cd="arith1" name="times"/>
            <OMS cd="nums1" name="i"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>real</Name>
<Role>application</Role>
<Description> 
This represents the real part of a complex number
</Description> 
<CMP>
 for all x,y | x = real(x+iy)
</CMP>
<FMP>
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      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="x"/>
        <OMA>
          <OMS name="real" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>imaginary</Name>
<Role>application</Role>
<Description> 
This represents the imaginary part of a complex number
</Description> 
<CMP>
for all x,y | y = imaginary(x+iy)
</CMP>
<FMP>
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
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        <OMV name="x"/>
        <OMV name="y"/>
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      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="y"/>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>complex_polar</Name>
<Role>application</Role>
<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).
</Description>

<CMP>
for all r,a | complex_polar(r,a) = r*e^(a*i)
</CMP>

<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
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    <OMBVAR>
      <OMV name="r"/>
      <OMV name="a"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMV name="r"/>
	<OMA>
	  <OMS cd="transc1" name="exp"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMV name="a"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>

<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)
</CMP>

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      <OMA>
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        <OMA>
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	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="y"/>
      </OMA>
      <OMA>
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        <OMA>
          <OMS cd="arith1" name="times"/>
	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ></FMP>

<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) </CMP>

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      <OMA>
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      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMA>
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	  <OMV name="a"/>
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	    <OMS cd="arith1" name="times"/>
	    <OMI> 2 </OMI>
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	    <OMV name="k"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
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</OMOBJ></FMP>

<Example> i = complex_polar(1,pi/2) 
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  <OMA>
    <OMS cd="complex1" name="complex_polar"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="nums1" name="pi"/>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMA></OMOBJ>
</Example>
</CDDefinition>

<CDDefinition>
<Name>argument</Name>
<Role>application</Role>
<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.
</Description>

<CMP> for all r,a | argument(complex_polar(r,a)=a) </CMP>

<FMP>
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      <OMA>
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        <OMA>
          <OMS cd="complex1" name="complex_polar"/>
          <OMV name="r"/>
          <OMV name="a"/>
        </OMA>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>

<CMP> the argument of x+i*y = arctan(y/x) (if x is positive) </CMP>

<FMP>
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          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="arctan"/>
        <OMA>
          <OMS cd="arith1" name="divide"/>
	  <OMV name="y"/>
	  <OMV name="x"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>

<CMP>
  the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2)
</CMP>

<FMP>
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        <OMV name="y"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="transc2" name="arctan"/>
      <OMV name="y"/>
      <OMV name="x"/>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>conjugate</Name>
<Role>application</Role>
<Description> 
A unary operator representing the complex conjugate of its argument.
</Description>
<CMP>
 if a is a complex number then (conjugate(a) + a) is a real number
</CMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
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      <OMV name="a"/>
      <OMS cd="setname1" name="C"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMA>
          <OMS cd="complex1" name="conjugate"/>
          <OMV name="a"/>
        </OMA>
        <OMV name="a"/>
      </OMA>
      <OMS cd="setname1" name="R"/>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>
</CDDefinition>

</CD>