1. (а) Using an appropriate contour in the upper half plane, find the integral z-1 dz....
2 +1 (b) Evaluate the contour integral dz, 22 – 9 where I is the boundary of the square D = {z E C:-4 < Re(z) < 4, -4 < Im(z) < 4} traversed once counterclockwise.
Exercise 10.12. i) (A contour integral) Let TR -R, RSR, where R E (1, 0o) half-plane with centre 0 that goes and SR is the semicircular arc in the upper from R to-R. Use a clear sketch of「R and the calculus of residues to evaluate the integral 「R 1+24 , and then decide which of the following five statements is true: (A) Rel < -3 (B) ReI e [-3,-1); (C) ReI [-1,1) (D) ReI E [1, 3); (ii) (An integral...
1. Evaluate the complex integral: ∫C [zRe(z) − z¯Im(z)]dz, where C is the line segment joining −1 to i. (z¯ = z bar) 2. Evaluate the complex integral: ∫ C [iz^2 − z − 3i]dz, where C is the quarter circle with centre the origin which joins −1 to i.
Show that the following integral has the solution, using contour integration in the complex plane: 2.π dx 9-V 3 Determine a fitting path of integration Hint: Show that the sum of the residue/residues of the singular point/points of the 2.T-1 integrand is and use (if appropriate) finally that sin3 -2 to calculate the integral.
Show that the following integral has the solution, using contour integration in the complex plane: 2.π dx 9-V 3 Determine a fitting path of integration Hint:...
1. Let P(x) = 22020 – 3:2019 + 22 -3. (b) Compute the contour integral Scof(z)dz with f(z) := 2 fled with f(-) -- 2021 – 222020+2 P2) +, where C (0) is the circle 121 = 8 with positive orientation.
1. (20 points) Let C be any contour from z = -i to z = i, which has positive real part except at its end points. Then, consider the following branch of the power function zi+l; f(3) = 2l+i (1=> 0, < arg z < Now, evaluate the integral Sc f(z)dz as follows: (a) (5 points) First, explain why f(z) does not have an antiderivative on C, but why the integral can still be evaluated. (b) (5 points) Then, find...
the previous hw question and answer
1. Consider the integral from question 2 of the previous homework assignment: too sin ma dx, and assume that both m and a are positive real numbers. By using an indented contour, evaluate this integral fully. You are allowed to resubmit material submitted as part of the previous assignment if you wish.] 2. (30 marks] Evaluate the following integrals: too sin ma x(x2 + a2) dar, m, a real, a +0. rt eike dx,...
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Ouestion 2. 2/2 -Throughout this question, z E C \ R and we define do (a) Locate and classify all singularities in the complex plane of Determine any associated residues (b) Evaluate Φ(z) by completing the contour in the upper half-plane. (c) Evaluate Ф(z) by completing the contour in the lower half-plane. (d) Verify that your answers to (b) and (c) are the same. (e) If r e...
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0 slit along the segment from 0 to i, a nonpolygonal region. (Use the principal square root throughout.) Hint: The desired non-polygonal region can be obtained as a "limit" of a sequence of polygonal regions.)
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0...