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...
[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)...
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
Let p(z) be the principal branch of 21-i. Let D* = C\(-00,0) be all the complex numbers except for the non-positive real numbers. (a) (4 points) Find a function which is an antiderivative of p(x) on D". (b) (6 points) Let I be a contour such that (i) I is contained in D* and (ii) the initial point of I' is 1 and the terminal point of I is i. Compute (2)dr. Justify your answers.
2. Let f(z) be the principal branch of i.е., f(z) exp@ Log(z)}. Co mpute (e)dz where C is the semicircle {et : 0 < θ < π
I need help with #5. Use the technique of Example 2 to determine the number of zeros of f in the first quadrant. 6. f(2)z6237 Example 2 Find the number of zeros of the function f(z) 23-222 4 in the first quadrant. Solution We examine f(z) on the contour shown in Figure 3.3; R is very big. On the segment 0 x R,f(x) = x3-2x2 + 4 is real and greater than 2. On the quarter- circle z = Re",...
,1/2 wi Problem (2) For the usual (branch) choice, namely -r < 0 < r let f(z) = positive. Compute Res/(a) a real and ,1/2 wi Problem (2) For the usual (branch) choice, namely -r
Consider the function f(t)=10sec^2(t)−3t^3 Let F(t) be the antiderivative of f(t) with F(0)=0. Then F(t)=...
5. Let f(x) = e-} Log(z) (that is, f is the principal branch of z-1/2). Compute [flade, where (a) (2 points) y is the upper half of the unit circle C(0) from +1 to -1; (b) (2 points) y is the lower half of the unit circle C1(0) from +1 to -1.
1. (20pts=7+5+8) (a) Find the order of the zero z = 0 of the function f(3) = ** (e*- 1). (b) Let 2 denote the principal branch of z3. Can in power of z in the annular domain be expanded in Laurent's series ann (0;0, R) = {2 € C:0< |2|< R} for some R >0? (c) Find the Laurent series in powers of 2 (i.e., Zo=0) that represents the function f(3) = in the annular domain 1 < 121...