sb3-hw03-2016-key

# sb3-hw03-2016-key - JHU 580.429 SB3 HW3 1 Evaluate dz 1z...

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JHU 580.429 SB3 HW3 1. Evaluate H dz 1 z for the following closed contours expressed in polar coordinates, z = re i θ with r > 0. (a) A single counter-clockwise loop, θ = 0 to 2 π . 2 π i (b) A single clockwise loop, θ = 2 π to 0. - 2 π i (c) A double counter-clockwise loop, θ = 0 to 4 π . 4 π i (d) A complicated path with m counter-clockwise loops and n clockwise loops. ( m - n ) 2 π i 2. Evaluate I ds 2 π i e st ( s + a )( s + b ) over the following closed contours, each a single counter-clockwise loop, with a > b > 0. (a) A contour including - a but excluding - b . I ds 2 π i e - at e ( s + a ) t ( s + a )( s + b ) = e - at - a + b I ds 2 π i 1 s + a = e - at b - a (b) A contour including - b but excluding - a . I ds 2 π i e - bt e ( s + b ) t ( s + a )( s + b ) = e - bt - b + a I ds 2 π i 1 s + b = e - bt a - b (c) A contour up the imaginary axis and then closed in the left half-plane: z = - i + i → - + i → - - i → - i . e - at - e - bt b - a Version: 2014/09/21 15:35:44 1 of ??

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JHU 580.429 SB3 HW3 (d) A contour up the imaginary axis and then closed in the right half-plane: z = - i + i + i - i
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