The circuit C, depicted below, is traversed clockwise starting and ending at -1. It consist of two parts: C = A + B where A c {z = z + iy : y = 1-2 and (a) Give parametrisations of A and B (b) Evaluate the line integrals L,-/ Re(z) dz, L,-, Re(z) dz. 1+2 (1+z+ z2)2 Calculus for complex line integrals to evaluate (c) Let f be given by f(z) - 2% Use the Fundamental Theorem of (d) Does...
Consider the function z(z-3) f (z) = - (z+1)2 (22+16) Syntax notes: • When entering lists in the questions below, use commas to separate elements of the list. Order does not matter. • The complex number i is entered as I (capital i). (a) List all the poles of f(z). -1,4-1,-4*1 BD (b) Enter the residue of the second-order pole. -1/4 OD
Evaluate the integral. Does Cauchy's theorem apply?
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2 & de 1 6 z dz > ¿ z2+ CZ til: i Z2+1 C: 12-11 Counterclockwile Counter clock wise
1. Consider the definite integralZcoswhere lal 2π 0 1+a2-2acos(x) a. (1 point) Using the change of variable z el, convert the real integral to a line integral over the unit circle in the counter-clockwise direction on the complex plane, i.e, f (2)dz. b. 1 point) Find the poles for f(z) in part a (use quadratic formula). What order are each of the poles? (1 point) Find the residues corresponding to each pole. (2 points) Using the residues in part c,...
QUESTION 2.
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52 Complex Analysis Exercises (1) Does the function w = f(2) za have an antiderivative on C? Explain your answer. (2) Is (z dz = 0 for every closed contour I in C? How do you reconcile your conclusion with Cauchy's integral theorem? (3) Compute fc Log(x+3) dz, where is the circle with radius 2. cente at the origin and oriented once in the counterclockwise direction. (4) Let I...
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.
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
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.
Question 3 [25 points]: Complex integration Subquestions (a), (b), and (c) will use C1 shown in the figure on the left-hand side, whereas subquestion (d) will use C2 shown in the figure on the right-hand side. Im (2) Im (2) SA= 1 → Re (2) → Re (2) 20 = 1 - (a) [3 points) Find a parametric representation for the curve Ci. (b) [7 points] Compute the integral Sc, z dz. (c) [5 points) Compute the integral Se, 22...
2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j.
2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j.