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6 Nov 2019
This problem deals with the evaluation of t he improper real integral As we saw in class, it. is sometimes possible to find the value of such integrals by a) using a suitable complex function and b) selecting an appropriate contour. For this problem, consider t.be contour pictured below: The horizontal line (containing -R. -r, r, and R) is the real axis in the complex plane, while Sr and Sn are upper semi-circles as in Problems 2 and 3. Show thatLet C be t.hc contour bounded by SR, Sr, and the real line segments from -R to -r and r to R, directed as shown above, and let f(z) = etz/z. Use Cauchy's Integral Theorem to compute f(z) dz and explain why f(z)dz + f(z) dz= f(z) dz- f(z) dz. Use part (b) and the rcsult.s from Questions 2 and 3 to determine the value of I. Show transcribed image text
This problem deals with the evaluation of t he improper real integral As we saw in class, it. is sometimes possible to find the value of such integrals by a) using a suitable complex function and b) selecting an appropriate contour. For this problem, consider t.be contour pictured below: The horizontal line (containing -R. -r, r, and R) is the real axis in the complex plane, while Sr and Sn are upper semi-circles as in Problems 2 and 3. Show thatLet C be t.hc contour bounded by SR, Sr, and the real line segments from -R to -r and r to R, directed as shown above, and let f(z) = etz/z. Use Cauchy's Integral Theorem to compute f(z) dz and explain why f(z)dz + f(z) dz= f(z) dz- f(z) dz. Use part (b) and the rcsult.s from Questions 2 and 3 to determine the value of I.
Show transcribed image text