Unformatted text preview: I (a) := Z ∞ √ x d x (1 + x ) 2 and I (b) := Z ∞ x 1 / 3 d x (1 + x )(2 + x ) by relating each to a contour integral around an appropriate “keyhole contour.” 4. (Problem 14.7.42, p. 701) Let F ( z ) := f ( z ) /f ( z ), where f ( z ) is analytic except at isolated points. a. Show that the residue of F ( z ) at an n th-order zero of f ( z ) is n . Hint : If f ( z ) has a pole of order n at a , then f ( z ) = a n ( z-a ) n + a n +1 ( z-a ) n +1 + ··· . b. Show that the residue of F ( z ) at a p th-order pole of f ( z ) is n . Hint : See the deﬁnition of a pole of order p at the end of Section 14.4....
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- Fall '10
- Definite Integrals, Methods of contour integration, Florida Atlantic University