pset7

# pset7 - I(a:= Z ∞ √ x d x(1 x 2 and I(b:= Z ∞ x 1 3 d...

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Physics (PHZ) 3113 Mathematical Physics Florida Atlantic University Fall, 2010 Problem Set VII Due: Tuesday, 16 November 2010 1. (Problems 14.7.6, 12 and 16, p. 699) Evaluate the following deﬁnite integrals of real functions by relating them to contour integrals in the complex plane. I (a) := Z 2 π 0 d θ (2 + cos θ ) 2 , I (b) := Z 0 x 2 d x x 4 + 16 and I (c) := Z 0 x sin x d x 9 x 2 + 4 . 2. (Problems 14.7.25 and 27, p. 700) Consider the deﬁnite integrals I (a) := Z 0 x sin x 9 x 2 - π 2 d x and I (b) := Z 0 cos πx 1 - 4 x 2 d x. Determine whether each integral exists. If it does, evaluate it by relating it to a contour integral. If not, then evaluate its Cauchy principal value by relating that integral to a contour integral. 3. (Problems 14.7.34 and 35, p. 700) Evaluate the deﬁnite integrals
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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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