1. (20pts=7+5+8) (a) Find the order of the zero z = 0 of the function f(3)...
9. Find the Laurent series about 0 that represents the complex function f(z)22 sin in the domain 0 < Izl < 00 0o rn i+ Answer: 9. Find the Laurent series about 0 that represents the complex function f(z)22 sin in the domain 0
Do Task 212 Task 211 (C). Find the Laurent series of exp z exp-, and exp-2 at zo = 0. From the definition of the coefficients for the Laurent series off at zo, we see that a-1 = Res(f, zo). Sometimes it is easier to find the Laurent series than the residue directly Task 212 (C). Using the results of Task 211, find Res (exp 1,0), Res(-exp z,0), and Res(exp "In fact, given a function f(z) that is holomorphic on...
2 7. Find the Laurent series of the function f(2) = in the region 1 < 121 < 2. (z+1)(2 – 2)
Derive the Laurent series expansion for the function (a) f(z) := z^2 sin (1/(z − 1 )) on the exterior |z − 1| > 0 of the unit disk centered at 1, and for the function (b) g(z) := 1 /(z^2 + z − 2) in the annular region 1 < |z − 1| < 3
) 1. Find the Laurent series of f(z) on the indicated domain. (a) -,2, on 0 < |z-i| < 2. 1+22 222z 5 , on z 1| > 1
2. Find three different Laurent series representations (about 0) for the function 3 f(z) 2. Find three different Laurent series representations (about 0) for the function 3 f(z)
Complex Analysis: = Define the function 22 f(z) 22 +1 For each annulus region given below, find the Laurent series of f(z) convergent in the region. (a) 0 < 12 – il < 2 (b) 1 < 121.
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:
Find Laurent series expansion centered on z= 0 for |z|<1 and for |z|>1 f(e) f() = -1-2) 1+22
(b) Determine the largest R such that the Laurent series of 2+1 f(z) = z 1 + (z-iT)(z+T) converges for 0< z-1|<R (b) Determine the largest R such that the Laurent series of 2+1 f(z) = z 1 + (z-iT)(z+T) converges for 0