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Let p(z) be the principal branch of 21-i. Let D* = C\(-00,0) be all the complex...
[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)...
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:
3. Let f(z) = zc where c is a complex number. Assume that the domain of f is the whole complex plane except the negative real numbers. a) What is the derivative of f? b) Let g(z) = cz. Find the derivative of g. 3. Let f(z) = zº where c is a complex number. Assume that the domain of f is the whole complex plane except the negative real numbers. a) What is the derivative of f? b) Let...
Complex 6. Find a branch of the multiple-valued expression f(z)-log(i(z + 21)) which is analytic for all z in the open disk
Let z and w be non-zero complex numbers such that zw /=1. Prove that if z= z^(-1) and w=w^(-1),then (z + w)/(1+ zw) is real.I know z * z^(-1) = 1.
Question 4. (a) Let c be a cluster point of a set S. Prove directly from the e, o definition of continuity that the complex valued function f() is continuous within S at the point c if and only if both of the functions Re[f(a) and Im[f(2)] are continuous within S at the point c (b) For which complex values of (if any) do the following sequences converge as n → oo (give the limits when they do) and for...
2. Problem 2 Let g(z) be a differentiable function defined on is shown below. Also suppose that g(2)-3 realnumbers. The graph of its derivative, g'(z), g'(a) Also define the differentiable, odd function hz) on all real numbers. Some values of h(z) are given below 0 12 3 4 5 h(z 02-42 2 (a) Calculate each of the following quantities or, if there isn't enough information, explain why i. (g'(x) +2) dr i.h() da ii. (h'(z) +2z) dr iv. 8h(x) dr...
Using Complex Numbers Theorems 3. (a) Let A= {()": n e z} and let {1, 2 1+1 1-1 -1+i B 1, -1, i, -i, V2 V2 V2 V2 What can you say about A and B? Is A CB, or B C A, or neither? If one is a subset of the other, is it a proper subset or not? Justify your answer. (b) Let D = {li :{w ": n € z}. What elements does this set consist of?
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation: (I point) Let F=21+(z...