Manipulating the Forward and Reverse Fourier Transform by Chrstopher Lusardi These are all the things that I found on complex number manipulation. Question: How does one manipulate the Forward and Reverse Fourier Transform? My best reference is: Introduction to Complex Variables by F. Greenleaf (QA 331 G73). ----------------------------------------------------------------------------- z = a + bi polar form: r(cos theta + i sin theta) exponential form: r exp(i theta) cos theta = a/square-root(square(a) + square(b)) sin theta = b/square-root(square(a) + square(b)) [Notation: exp (i) = e to power i] ----------------------------------------------------------------------------- n (cos theta - i sin theta) = cos n theta + i sin n theta ----------------------------------------------------------------------------- cos theta = exp(i theta) + exp(-i theta) n ---------------------------- 2 sin theta = exp(i theta) - exp(-i theta) ---------------------------- 2 i ----------------------------------------------------------------------------- f(x) = exp(i x) = cos x + i sin x this can be written: f(x) = u(x) + i v(x) derivative of f(x) is: f'(x) = u'(x) + i v'(x) similarly: integral f(x) dx = integral u(x) dx + i integral v(x) dx ----------------------------------------------------------------------------- d 2 -- (z + z + 1) = 2z + 1 dz ----------------------------------------------------------------------------- b b integral (alpha times f)(t) dt = alpha integral f(t) dt a a if alpha is any complex number ----------------------------------------------------------------------------- d -- (exp(alpha x)) = alpha exp(alpha x) if alpha is any complex number dx integral exp(alpha x) dx = 1/2 exp(alpha x) + Constant alpha not equal 0 ----------------------------------------------------------------------------- exp (i theta) = 1 iff theta = 2 pi k ----------------------------------------------------------------------------- z = r exp(i theta) and w = p exp(i phi) are equal iff r = p and theta - phi = 2 pi k ----------------------------------------------------------------------------- |a + i b| = square-root(square(a) + square(b)) ----------------------------------------------------------------------------- exp(i pi) = cos pi + i sin pi = -1 exp(z + 2 pi i) = exp (z) exp(2 pi i) = cos 2 pi + i sin 2 pi = 1 n (exp(2 pi i/n)) = exp(2 pi i) = 1 Table: ............................................................................. |||theta = 0||theta = pi/4 || theta=pi/2||theta = pi|| theta = 3 pi/2 ............................................................................ exp(i theta) = ||| 1 || 1/square-root(2)(1 + i)||i ||-1 || -i ............................................................................ exp(x + i y) = exp(x)(cos y + i sin y) = (exp(x) cos y) + i(exp(x) sin y) ----------------------------------------------------------------------------- exp(z + w) = exp(z) exp(w) exp(-z) = 1/exp(z) exp(w - z) = exp (w)/exp(z) exp(-i theta) = 1/exp(i theta) exp(i theta) = 1 iff theta = 2 pi k, for k = ... -2,-1,0,1,2,... ----------------------------------------------------------------------------- e(z) is periodic when the variable is translated by 2 pi i: exp(z + 2 pi i) = exp(z) forall z ----------------------------------------------------------------------------- A complex number Wo is called a period for a function f(z) if we have f(z + Wo) = f(z) forall z; The numbers ... -2Wo,-Wo,0,Wo,2Wo,... are also periods. The numbers Wk = 2 pi k i are periods for exponential function. The exponential function has no other periods. This periodicity of exp(z) means that we know how exp(z) behaves everywhere in the plane once we know how it acts in any horizontal strip of width 2 pi: Eg: i y ^ 4 pi-|<. | . |. 2 pi-|< 0 | . <-----------------------|<--------------------> x | . -2 pi-|. | | V ----------------------------------------------------------------------------- exp(u + i v) = exp(u) exp(i v) cos(z) has periods Zk = 2 pi k + i theta tan(z) has periods Zk = pi k + i theta tan z = sin z/cos z ----------------------------------------------------------------------------- -- | .-, ###|Christopher M. Lusardi, Richmond Quadrangle | / / __ , _ ###|University At Buffalo, Buffalo New York | \_>/ >_/ (_/\_/<>_ |library catalog + abstracts: bison.acsu.buffalo.edu |_ 14261 _|(When in doubt ask: xarchie, xgopher, or xwais.) Discuss this article in the forums Date this article was posted to GameDev.net: 7/16/1999 (Note that this date does not necessarily correspond to the date the article was written) See Also: Fourier Transforms © 1999-2011 Gamedev.net. All rights reserved. Terms of Use Privacy Policy Comments? Questions? Feedback? Click here!