1. Use an antiderivative to show that for every contour C extending from a point zi...
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...
Problem 5: Let f(z) = zi = eiLog?, [2] > 0, -T < Arg z <a denote the principal branch of the function z', and let C be any contour from –2 to 1 that, except for its endpoints, lies above the real axis. (a) Find an antiderivative of the function f(z); (b) Compute the integralf(z)dz; SOLUTION:
9. Evaluate o (x+ 1)2 by extending the integral into the complex plane and using the contour C shown below. 9. Evaluate o (x+ 1)2 by extending the integral into the complex plane and using the contour C shown below.
QUESTION 2. PLEASE USE COMPUTER WRITING SO I CAN READ IT 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...
Show that . A question from (mathematical physics - Couchy integral formula) c) The Rodrigues formula of Legendre polynomials can be converted into the Schlafli integral as (-1)" 1 (1 - 22n Pn(x) = dz 2n 2ni (z - x)n+1 C is a closed contour encircles the point z = x C
Problem 6. (1 point) Use the contour diagram of f in the ligure below to decide ir the speciñied directional derivatives below are positive, negative, or approxmately zero 14 (a) At point (-2,2). in direction-i. is.? (b) At point (0,-2) in direction- i f s ? (c) At point ( 1,1), in direction i + s ? a) At point (-1,1), in direction +j f: s ? (e) At point (0,-2), in direction it2j. fd is!? n At point (0,-2),...
Show that integral dz/(z-1-i)n+1 =0, if n does not equal 0 and 2 pi i if n = 0 for C the boundary of the square 0<=x<=2, 0<=y<=2, taken counterclockwise. [Hint: Use the fact that contours can be deformed into simpler shapes (like a circle) as long as the integrand is analytic in the region between them. After picking a simpler contour, integrate using parametrization.]
Problem 3. (i) Show that the Taylor series expansion of the function , with center at 1, is for -1<1 ii) Explain why the function Log z is analytic in the disk l:-1 iii) For each point z with :-1< 1 consider the straight line segment C starting at 1 and ending at z. Evaluate dz. Hint: You do not need to do any computation. Note that Logz is an antiderivative of 1/z in the disk :-1<1.) (iv) Integrate each...
. (a) Show that the function u= 4x2 - 12.xy2 is harmonic and v=12.xy-4v2 is a harmonic conjugate of u. [Consequently, the function f =u+iv is entire, thus it has an antiderivative and that any contour integral of f is path independent.] (b) Find an antiderivative F(-)= F(x+iy)=P(x, y)+i Q(x, y) of the function f; and (c) evaluate ( f (2) ds , where C is any contour from 0 to 1–2i .
EXERCISE 6 Let Zi, Z2,-.., Zi6 be an i.i.d. sample of size 16 from the standard normal distribution N (0,1). Let Xi,X2,..., X64 be an i.i.d. sample of size 64 from the normal distribution (μ, 1). The two samples are independent. a. What is the distribution of Y, where Y-Σ161 Z2 + Σ-i(X-μ)2? Study List b. Find ΕΥ. c. Find Var(Y) d. Approximate P(Y105)