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Midterm 1 Solutions

# 3 usipg the calculus of residues evaluate the

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Unformatted text preview: ot;. 3. Usipg the calculus of residues,. evaluate the integral " S 21l' cos {ne- sinlJ}. exp (coslJ) de o for all integral values of n. 4. Show that, for -1 < P < 1 and -Jt < A < Jt, the integral 00 1 X 1+ 2 x. f tAx c.er:r 5;•.. A 71 .x + xl. si", Hint: it will be helpful to write cos)., as Iff .;i- ( e i. A + f .s;1'\ ~ (Euler) e" (. .A ). .. .. 5. By integrating a suitable function around a closed contour in the z-plane, evaluate the integral SihX x( X 1- + 6.. 2. ) dx ) a being a real number. 6. Evaluate the Inverse Laplace Transform of the function .... f) (\$ = s / (s 4 - a 4 ) ~; and check the limiting case a -0. ~;' / Cf)'\$ 'J-)( J~t 10 1.t ()< ) 'I)} W...
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