Fourier Transforms

CHAPTER 7 COMPLEX FUNCTION THEORY

   

Up to this point we have worked exclusively with real- or complex-valued functions of real or integer variables, for example,

• f(t) = e^{− t}U(t), a real-valued function of the real variable t;
• , a complex-valued function of the real variable ν; and
• f[n] = e^iθn, a complex-valued function of the integer variable n.

A complex-valued function may be separated into its real and imaginary parts, . Once this is done, it can be manipulated according to the methods of ordinary real calculus, applied individually to the real and imaginary parts, for example,

The subject of the next two chapters is functions of a complex variable. The motivation for this study is twofold. First, there are insights and methods from complex analysis applicable to solving Fourier transform problems. Second, there are other useful transforms, related to the Fourier transform, whose use requires some facility with complex variable theory. These are the Laplace transform,

and its discrete-time counterpart, the Z transform,

The transform variables s and z are complex, ...