Chapter 5 - Functions of a complex variable
Published online by Cambridge University Press: 05 June 2012
Summary
In Chapter 1 we introduced some particular functions of a complex variable, such as powers and the exponential, that were needed for the chapters to follow. We now take up the subject again and develop a general theory of functions of a complex variable, including integration in the complex plane. Since the plane has no physical meaning it might seem that we are embarking on a study that has no relevance to an engineer, but a familiarity with complex-variable techniques is important in all branches of engineering. In particular, they are essential for a proper appreciation of the Fourier- and inverse Laplace-transform operations discussed in later chapters, and this is the reason for their consideration now.
General properties
We consider complex functions of the complex variable z = x + iy, where the definition of a function is analogous to that in real variable theory:
If w and z are any two complex numbers, then w is a function of z (i.e., w = f(z)) if, to every value of z in some domain D, there corresponds one or more value(s) of w.
A domain is simply an open region of the complex plane bounded by a closed contour, and we shall use such terms with no more precise definition than geometrical intuition requires.
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- Information
- Mathematical Methods in Electrical Engineering , pp. 133 - 204Publisher: Cambridge University PressPrint publication year: 1986