1. Let P(x) = 22020 – 3:2019 + 22 -3. (b) Compute the contour integral Scof(z)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.
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1 Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
2. (a) Evaluate the contour integral z dz, where I is the circle 12 – 11 = 2 traversed once counterclockwise.
the other pic was wrong. this are the ones i need help these are my questions it was croped wrong before 2. Let C denote the circle \z - il = 3 and compute f(z)dz if (a) f(x) = 332 (b) f(z) = + C) f(z) = sin(22) (d) f(x) = 212 3. Let f(x) = zz_42+10 and compute Sf(z)dz if C is the contour (a) Iz| = 1 (b) 121 = 4 (c) |z| = 7 2. Let C...
[3] Let p(z) be the principal branch of 21-1. Let D* = C\(-0,0] be all the complex numbers except for the non-positive real numbers. (a) Find a function which is an antiderivative of p(z) on D*. (b)Let I be a contour such that (i) T is contained in D* and (ii) the initial point of is 1 and the terminal point of I is i. Compute J, Plzydz. Justify your answers. [9] Let f(z) be the function 2 3 f(x)...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...
Exercice 1 We consider the function f(x) = 2 #0 and for r > 0. let S, = {€ C/2 = r} with positive orientation. For 0 < <R, we denote by r the curve consisting of SRUT-R,-€) US, UL, R), where S = {z E C/121 = } with negative orientation. 1. Prove that o = [513)dz = [5(=)dz + [s()de – [ (dz + 1" $(x)dr.
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...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top hemisphere of the unit sphere oriented upward, and C the unit circle in the xy-plane with positive orientation. (a) Compute div(F) and curl(F). (b) Is F conservative? Briefly explain. (c) Use Stokes’ Theorem to compute ? F · dr by converting it to a surface integral. (The integral is easy if C you set it up correctly) 4. (23 pts)...
1. (а) Using an appropriate contour in the upper half plane, find the integral z-1 dz. (z - i)(z+3i)2 If the contour was closed in the lower half plane, explain how your (b) residue calculation would change.