Integration in the Complex Plane
In order to evaluate the intractable integrals arising from Equation 4.53, we need to devote some space first to the theory of functions of a complex variable. We shall present with no proof, or very little, many results that deserve careful study and rigorous treatment. The reader is advised to read one of the good books on the subject,1 or to attend lectures on the subject by a mathematician.
KeywordsRiemann Surface Phase Angle Taylor Series Branch Point Complex Variable
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- 1.Among the well-known books on functions of a complex variable are: Churchill, R. V., J. W. Brown, and R F. Verhey. Complex Variables and Applications. New York; McGraw-Hill, 1974; Franklin, P. 3d ed. Functions of Complex Variables. Englewood Cliffs, NJ: Prentice-Hall, 1947.Google Scholar
- 2.The uniqueness of the derivative at a point must survive also the use of various complicated spirals along which h may approach zero. In fact, the C-R conditions and the continuity of the functions u and v are sufficient to guarantee uniqueness. See Churchill, R V., and J. W. Brown. Fourier Series and Boundary Value Problems 3d ed. New York: McGraw-Hill, 1978.Google Scholar