homework8 - and whether we should take the semi-circle in...

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Due 03-13-09 Physics 132 Winter 2009 P. Kraus HOMEWORK #7 Hand into box marked “132” outside 1-707D PAB by noon Fri. —————————————————————————————————————————— 1) BC p. 267 problems 1,3, 5: Use residues to evaluate the following integrals Z 0 dx x 2 + 1 , Z 0 dx x 4 + 1 , Z 0 x 2 dx ( x 2 + 9)( x 2 + 4) 2 Ans: π/ 2 , π/ (2 2) , π/ 200. 2) An integral R -∞ f ( x ) dx can be computed efficiently via residues if we can close the contour by adding in a semi-circular contour in the upper or lower half complex plane. This leaves the integral unchanged provided that the semi-circle contour gives a vanishing contribution to the integral in the R → ∞ limit. For the functions f ( x ) below, explain whether this is the case
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Unformatted text preview: and whether we should take the semi-circle in the upper half or lower half plane (or whether either is allowed). f ( x ) = a ) x 2 + 3 ( x 4 + 9) , b ) x 2 ( x + 1)( x + 2)( x + 3) , c ) e-2 ix x 2 + 13 , d ) xe 3 ix x 2 + 13 , e ) x 2 e 3 ix x 2 + 13 3) BC p. 267 problem 8: Use a residue and the contour. .. 4) BC p. 275-276 problems 1, 6, 10: Use residues to evaluate the following (assume a,b > 0) Z - cos xdx ( x 2 + a 2 )( x 2 + b 2 ) , Z - x 3 sin ax x 4 + 4 dx , Z - ( x + 1) cos x x 2 + 4 x + 5 dx Ans: a 2-b 2 ( e-b b-e-a a ) , e-a cos a, e (sin 2-cos 2). 5) BC p. 287 problem 6: Show that. .. 1...
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