s7_95-100a

s7_95-100a - √ π 2 2(30 pts Evaluate the following...

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ACM95a/100a November 18, 2009 Problem Set VII Please deposit your completed homework set to by the due-time in the Firestone 303 door-slot. Please write your Grading Section Number at the top of the ±rst page of your homework set. 1. (20 pts.) Integrate exp( - z 2 ) around the boundary of the sector | z | < R , 0 < θ < π/ 4 and deduce that i 0 sin( t 2 ) dt = i 0 cos( t 2 ) dt = 1 2 i 0 e t 2 dt. [Hint: On the curved part of the contour | exp( - z 2 ) | = exp( - r 2 cos 2 θ ). Show analytically that cos 2 θ 1 - 4 π θ, 0 θ π 4 . This makes it easy to estimate the integral by a calculation similar to that in the proof of Jordan’s lemma.] The Frst two integrals are known as Fresnel integrals. Even though the integrands don’t decrease with increasing t , the integrals are Fnite because of the increasingly rapid oscillation of the integrands as t → ∞ . The latter integral is known from Calculus to equal
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Unformatted text preview: √ π/ 2. 2. (30 pts.) Evaluate the following integrals using contour integration: (a) (10 pts.) i ∞ dx (1 + x 2 ) 2 , (b) (20 pts.) i ∞ dx 1 + x 3 . [Hint for (b): use the boundary of a 120 ◦ circular sector of large radius as integration contour] 3. (30 pts.) Show that (a) (15 pts.) i ∞ (ln x ) 2 1 + x 2 dx = π 3 8 , (b) (15 pts.) i ∞ ln x 1 + x 2 dx = 0 . [Hint: Obtain both integrals simultaneously by integrating some branch of (log z ) 2 / (1 + z 2 ) around a large semicircle indented at z = 0.] 4. (20 pts.) Evaluate i ∞ x − α 1 + x 4 dx for α real. What are the restrictions on real α for this integral to exist? Due at 5pm on Wed. November 25....
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