Improper Integrals of Rational Functions

Definition A.14 (Cauchy Principal Value of Integral) For real x, iff x) is a continuous function on —oo X +00, the improper integral cff x) over [—00, +00] is 

Theorem A.5 (Evaluation of the Cauchy Principal Value of an Integral) If a function f z) is analytic in a simply connected main D, except at a finite number of singular points zi. 2jt, iff x) = P x)/Q x) where P(x) and Q(x) are polynomials, Q x) has no zeros, and the degree ofP x) is at least two less than the degree of Q(x), and ifC is a simple positively oriented (counterclockwise) closed contour that lies in D, then 

Solution The function has two poles on the real axis located atx = a and x = —a, 

the integral along the real axis can be evaluated by evaluating the integrals along the contours that skirt the poles. 

2 Show that the exponential of the complex conjugate e is not an analytic function in any domain. 

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Figure A.7 Demonstration of integration for a domain with two poles on the real axis. A.3.2 Improper Integrals of Rational Functions...

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