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. complex1 http://www.openmath.org/cd http://www.openmath.org/cd/complex1.ocd 2006-03-30 2004-03-30 3 1 Author: OpenMath Consortium SourceURL: https://github.com/OpenMath/CDs official This CD is intended to be `compatible' with the MathML view of operations on and constructors for complex numbers. complex_cartesian application 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.) for all x,y | complex_cartesian(x,y) = x + iy real application This represents the real part of a complex number for all x,y | x = real(x+iy) imaginary application This represents the imaginary part of a complex number for all x,y | y = imaginary(x+iy) complex_polar application 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). for all r,a | complex_polar(r,a) = r*e^(a*i) for all x,y,r,a | (r sin a = y and r cos a = x) implies (complex_polar(r,a) = complex_cartesian(x,y) 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) 2 i = complex_polar(1,pi/2) 2 argument application 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. for all r,a | argument(complex_polar(r,a)=a) the argument of x+i*y = arctan(y/x) (if x is positive) the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2) conjugate application A unary operator representing the complex conjugate of its argument. if a is a complex number then (conjugate(a) + a) is a real number