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. Supp...
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
5. This problem outlines a "bar room" proof of Cauchy's Integral Formula Assume Ω is a simply connected domain. Let f(z) be holomorphic on and suppose zo E S2. We know and inside Ω f(3) cr 220 f(e) dz where C is a circle centered at zo with radius r. (a) Express z-zo + reit where r is given by C and t [0,2 ] and rewrite the above integral in polar form. (b) From (a) let r-0 in the...
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
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
Problem 10.7. Show that azn-e* has n roots with Izl < 1 if lal > e. Problem 10.8. Suppose that f is holomorphic inside and on a toy contour γ. Suppose f has no zeros on the contour γ and that zi, ,«n are the zeros of f inside γ, the order of being kj. Show that j-1 for any function g which is holomorphic inside and on γ. Problem 10.9. Let u(z,y)-zy(x2-уг). Find the maximum and minimum values of...
Suppose f(z) is a holomorphic function in a domain U, and z0 ∈ U. Prove that f has a zero of order m at z0 if and only if f(z) = g(z)(z − z0)^m, where g(z) is holomorphic in U and g(z0) not equal to 0. Please prove both directions of the if and only if statement and use series expansion to prove. We have not learned calculus of residues yet.
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
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
7. Prove that the function N4-i/z is conformal in its entire domain Ω-C\ {0}. Find the image wo of the point v3 i under this function, as well as the rotation angle and the stretching/contracting factor of tangent vectors at this point. Find the images Vi and V2 of the vectors vi-1-(1,0) and v2 (0,1) under this map, and check that the angle between the images is the same as the angle between vi and v2 7. Prove that the...