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complex1.omcd
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<OMOBJ xmlns:om="http://www.openmath.org/OpenMath"
xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="meta" name="CD"/>
<OMA>
<OMS cd="meta" name="CDComment"/>
<OMSTR>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.</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDName"/>
<OMSTR>complex1</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDBase"/>
<OMSTR>http://www.openmath.org/cd</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDURL"/>
<OMSTR>http://www.openmath.org/cd/complex1.ocd</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDReviewDate"/>
<OMSTR>2006-03-30</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDDate"/>
<OMSTR>2004-03-30</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDVersion"/>
<OMSTR>3</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDRevision"/>
<OMSTR>1</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDComment"/>
<OMSTR>Author: OpenMath Consortium SourceURL: https://github.com/OpenMath/CDs</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDStatus"/>
<OMSTR>official</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>This CD is intended to be `compatible' with the MathML view of operations on and constructors for complex numbers.</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>complex_cartesian</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>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.)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all x,y | complex_cartesian(x,y) = x + iy</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMS cd="relation1" name="eq"/>
<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>
</OMA>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>real</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>This represents the real part of a complex number</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all x,y | x = real(x+iy)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
</OMBVAR>
<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>
</OMA>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>imaginary</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>This represents the imaginary part of a complex number</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all x,y | y = imaginary(x+iy)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
</OMBVAR>
<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>
</OMA>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>complex_polar</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>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).</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all r,a | complex_polar(r,a) = r*e^(a*i)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<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>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all x,y,r,a | (r sin a = y and r cos a = x) implies (complex_polar(r,a) = complex_cartesian(x,y)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="r"/>
<OMV name="a"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="r"/>
<OMA>
<OMS cd="transc1" name="sin"/>
<OMV name="a"/>
</OMA>
</OMA>
<OMV name="y"/>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<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>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>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)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="set1" name="in"/>
<OMV name="a"/>
<OMS cd="setname1" name="R"/>
</OMA>
<OMA>
<OMS cd="set1" name="in"/>
<OMV name="k"/>
<OMS cd="setname1" name="Z"/>
</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>
<OMS cd="arith1" name="plus"/>
<OMV name="a"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMI> 2 </OMI>
<OMS cd="nums1" name="pi"/>
<OMV name="k"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMA>
<OMS cd="meta" name="Example"/>
<OMSTR>i = complex_polar(1,pi/2)</OMSTR>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMS cd="nums1" name="i"/>
<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>
</OMA>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>argument</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>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.</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>for all r,a | argument(complex_polar(r,a)=a)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="r"/>
<OMV name="a"/>
</OMBVAR>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="complex1" name="argument"/>
<OMA>
<OMS cd="complex1" name="complex_polar"/>
<OMV name="r"/>
<OMV name="a"/>
</OMA>
</OMA>
<OMV name="a"/>
</OMA>
</OMBIND>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>the argument of x+i*y = arctan(y/x) (if x is positive)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="relation1" name="gt"/>
<OMV name="x"/>
<OMS cd="alg1" name="zero"/>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="complex1" name="argument"/>
<OMA>
<OMS cd="complex1" name="complex_cartesian"/>
<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>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2)</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="complex1" name="argument"/>
<OMA>
<OMS cd="complex1" name="complex_cartesian"/>
<OMV name="x"/>
<OMV name="y"/>
</OMA>
</OMA>
<OMA>
<OMS cd="transc2" name="arctan"/>
<OMV name="y"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS cd="meta" name="CDDefinition"/>
<OMA>
<OMS cd="meta" name="Name"/>
<OMSTR>conjugate</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Role"/>
<OMSTR>application</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="Description"/>
<OMSTR>A unary operator representing the complex conjugate of its argument.</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="CMP"/>
<OMSTR>if a is a complex number then (conjugate(a) + a) is a real number</OMSTR>
</OMA>
<OMA>
<OMS cd="meta" name="FMP"/>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="set1" name="in"/>
<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>
</OMA>
</OMA>
</OMA>
</OMOBJ>