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
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
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...
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 that f : D[0, 1] → D[0, 1] is holomorphic, prove that |f 0 (z)| ≤ 1/(1 − |z|) ^2 for all z ∈ D[0, 1]. is there any way not to use the Schwartz' lemma
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.
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2
7. Suppose f: D→C is holomorphic. Show that the diameter d= sup,, wED f(a)-f(w)of the image of f satisfies Moreover, it can be shown that equality holds precisely when f is linear, f(z)- Note. In connection with this result, see the relationship between the diameter of a curve and Fourier seri . es described in Problem 1, Chapter 4, Book I -, Layer d' whenever 0 < r < İ.] Hint 25,(0) =亦 ici r
7. Suppose f: D→C is...
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...