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Chapter 7:
Complex Integration, Laurent Series, and Residues
In this chapter we extend the discussion of complex variables to complex integrals and Laurent series. Complex integrals in general behave similar to the line integrals found in vector analysis. However, the method of residue calculus used for evaluating integrals that involve an analytic function is more powerful than any counterpart in vector analysis—i.e., the residue calculus can be used to evaluate inverse Laplace transforms, and is also useful in calculating Fourier and Mellin transform integrals. Laurent series are a natural generalization of Taylor series that provide the basis for developing the residue calculus. The two-dimensional nature of a complex variable required us in Chapter 6 to generalize our notion of derivative in the complex plane. This was a consequence of the fact that a complex variable can approach its limit value from infinitely-many directions, rather than just two directions as in the case of a real variable. This two-dimensional aspect of a complex variable will also influence the theory of integration in the complex plane, requiring us to consider integrals along general curves in the plane rather than simply along segments of the x-axis. Because of this, we find that complex integrals behave more like line integrals from vector analysis instead of like standard Riemann integrals.
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