Complex Numbers and Functions

Fourier transforms of some simple funetions are given by Kreyszig [1], However, usually the Fourier transform is more eommonly expressed in complex form without specifying the nature of the funetion to be transformed. This type of Fourier transform is considered in the following seetion after a discussion of complex numbers and functions. 

Complex numbers are often used to deal with phenomena involving periodie functions. A complex number z is usually written as 

The square of j is —1, so that the product y y2 becomes part of the real term in Z1Z2. Division of one complex number by another illustrates the use of the complex conjugate of the denominator. Thus, 

Important complex functions are based on the Euler formula, that is, 

the Fourier transform can be written in complex notation. The Fourier transform of any function f(x) is written as 

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In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

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