MATH502_HW5 - Serge Ballif(1 Evaluate the integral MATH 502...

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Serge Ballif MATH 502 Homework 5 February 22, 2008 (1) Evaluate the integral Z -∞ x 2 cos x 4 + x 4 d x by means of contour integration. Be sure to justify carefully all the steps in your argument. To compute this integral, we will consider the function h ( z ) = z 2 e iz 4+ z 4 , which is bounded in the upper half plane and has real part equal to the integrand in our problem. We will integrate over the curve γ 1 and γ 2 as pictured below. R - R 1 + i - 1 + i γ 2 γ 1 Call the entire path Γ. Then by the residue theorem 1 2 πi Z Γ f ( z ) d z = Res( h, 1 + i ) + Res( h, - 1 + i ) . Since | e iz | ≤ 1 in the upper half plane and tends to 0 almost everywhere as R , we see that R γ 2 z 2 e z 4+ z 4 d z tends to 0 as R → ∞ by the dominated convergence theorem. Therefore, by the residue theorem Z Γ z 2 e z 4 + z 4 d z = Z γ 1 z 2 e z 4 + z 4 d z = 2 πi (Res( h, 1 + i ) + Res( h, - 1 + i )) . We are finished when we compute this, because R -∞ x 2 cos x 4+ x 4 d x = lim R →∞ R γ 1 z 2 e z 4+ z 4 d z To calculate the residues we note that z 4 + 4 = ( z - 1 - i )( z - 1 + i )( z + 1 - i )( z +1+ i ), so h ( z ) has simple poles at z = ± 1 ± i . We can compute the residues using the formula for simple poles: Res( h, 1 + i ) = (1+ i ) 2 e i (1+ i ) 4(1+ i ) 3 = (1 - i ) e - 1+ i / 8. Similarly Res( h, - 1 + i ) = ( - 1 - i ) e - 1 - i / 8. Thus Z -∞ x 3 cos x 4 + x 4 d x = Re 2 πi e - 1 ( ( e i - e - i ) + i ( - e i - e - i ) ) 8 !
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