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Essential Mathematical Methods for the Physical Sciences by K. F. Riley, M. P. Hobson

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15

Applications of complex variables

In Chapter 14, we developed the basic theory of the functions of a complex variable, z = x + iy, studied their analyticity (differentiability) properties and derived a number of results concerned with values of contour integrals in the complex plane. In this current chapter we will show how some of those results and properties can be exploited to tackle problems arising directly from physical situations or from apparently unrelated parts of mathematics.

In the former category will be the use of the differential properties of the real and imaginary parts of a function of a complex variable to solve problems involving Laplace’s equation in two dimensions, whilst an example of the latter might be the summation ...

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