2. Prove that if f(z) is analytic at oo, then it has a series expansion of the form an f(2)= n=0 converging uniformly o...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C? u(20) for all z...
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
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
(b) Let a >0. Does (f.) converge uniformly on [-a, al? (c) Does (f) converge uniformly on R? Q4 You are given the series n2 +r2 (a) Prove that the series converges uniformly on [-a, al for each a > 0. (b) Prove that the sum F(r) is well defined and continuous on R. (c) Prove that the series does not converge uniformly on R. Q5 You are given the series I n2r2 (b) Let a >0. Does (f.) converge...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on 3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on
Find Laurent series expansion centered on z= 0 for |z|<1 and for |z|>1 f(e) f() = -1-2) 1+22
8.3.1 Prove the following: (a) Every infinite series of the form 0O 72 (8.2) using appropriate choices for the coefficients Ibn], with the restrictioin that either bn- 0 or bn 1 is true for every n, represents a number in the interval [0, 1), with the exception that if bn n then the sum of the series is exactly 1. (b) Every real number in the interval [0,1) can be represented using a binary expansion, that is, can be represented...
part e and f 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series diverges. ak 1 + at ar ai ak 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series...
-a" (a) Find the Taylor series for sinx about x 0, and prove that it converges to sinx uniformly on any bounded interval [-N,N (b) Find the Taylor expansion of sinx about xt/6. Hence show how to annrmximate D. -a" (a) Find the Taylor series for sinx about x 0, and prove that it converges to sinx uniformly on any bounded interval [-N,N (b) Find the Taylor expansion of sinx about xt/6. Hence show how to annrmximate D.