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 comple...
) 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
Find a holomorphic function F(z) on Ω-{z I Izl < r} such that for any a E Ω, F(a) F(0)-Z dz. Suppose f(z) is entire and Ω is simply connected domain. Show lim 22-h2220 Find a holomorphic function F(z) on Ω-{z I Izl
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)
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
7. Find a holomorphic function F(z) on Ω-z I Izl < r} such that for any a E Ω, F(a) F(0)-da. 0 7. Find a holomorphic function F(z) on Ω-z I Izl
A) B) C) 1 Find the Laurent series for 22 +22 for 0 < 121 < 2 Find the Laurent series for (z+2)}(3-2) for 2 – 3) > 5 1 Find the Laurent series for z2(z-i) for 1 < 12 – 11 < V2
complex anaylsis f(2)= 22 2z 2²+1 For each annulus region below, find the Laurent series of fiz) convergent convergent in the region (i) OC/Z-il<2 12IZI
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.
Find Laurent series expansion centered on z= 0 for |z|<1 and for |z|>1 f(e) f() = -1-2) 1+22
(C)!!!!! 5. Find the Laurent series expansion of: 1 (a) f(x) = 1 about i, (b) f(x) = 22 + atz, convergent on {2< 121 < 4}, (c)* f(x) = 273-33+2, convergent on {{ < \z – 11 <1}.