Question

Evaluate the contour integral \oint \frac{e^{p z}}{1+e^{z}} d z where p is a constant, for the contour shown below (i.e., starting at R on the positive real axis, going to R+2xi, then to –R+2Ti, then to –R and back to R, all in straight line segments). [4] (b) Now assume that p is real and 0 < p < 1. As R → ∞, what is the contribution of the vertical segments of the contour (i.e., R to R+2ñi and-R+ 2ni to – R) to the contour integral in part (a)? [1] (c) Use the results of parts (a) and (b) to evaluate \int_{-\infty}^{\infty} \frac{e^{p x}}{1+e^{x}} d x for 0 < p< 1. Hint: Consider how the contribution of the upper horizontal contour (i.e., from R+2xi to –R+ 2ni) in part (a) relates to the integral in part (c). [5]

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