5. Let Ω be open in C and consider the set U in Ω that has no limit points in Ω. For the sake of your imagination, 0 could be the set of isolated zeros or poles of some mero- morphic function. Let C be a simple closed curve in Ω\U oriented counter clockwise. Can there exist infinitely many points of U contained inside the region bounded by C? Explain 5. Let Ω be open in C and consider the...
Please do 2 only please do 2 only Exercises (1) Compute for de and c ) da where is the ultime center at the origin and oriented once in the counterclockwise (2) Computer da, where I is the circle {: € C: 1:= 3) once in the counterclockwise direction (3) (Mean Value Property of Holomorphic Functions) Supposed w = f(e) is holomorphic on and inside the circle {: € C:- Prove that f(20) == f( 70 +re) de. (4) Under...
7. Let D = {z C z 1) denote the closed unit disc centered at the origin. Let f : D → C be a continuous function which is holomorphic on the interior of D. Suppose If(:) 2/(2- 2) and that If (z)1-2 for all z such that 1. Show that f(z is constant. 7. Let D = {z C z 1) denote the closed unit disc centered at the origin. Let f : D → C be a continuous...
12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3. 12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3.
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then there exists finitely many points yi ∈ E, i = 1,...,N such that Here Br(yi)(yi) is the open ball (neighborhood) of radius r(yi) centered at yi. Also, following definitions & theorems should help that E CUBy Definition. A subset S of a topological...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
4.8) a) Complete the statement of: The Divergence Theorem: Let D be a closed solid in space bounded by a closed surface s oriented by an outwardly directed unit normal vector n. If F(x, y, z)=(M(x,y,z), N(x, y, z), P(x, y, z)) where M, N, and P have continuous partial derivatives in D, then: D b) Use the Divergence Theorem to write as an iterated integral the flux of F=(x",x’y,x?:) over the closed cylindrical surface whose sides are defined by...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
8) This is essentially p.221, #15a), but using more clarified notation. Let D be a closed, bounded interval and f : D → R. Suppose that for each c E D there exists δ = and M = Mc both depending on c where If(x)| < M if |x-c| < δ and x E D. Prove that in fact f is bounded on D. That is, there exists M>0 with If (x)S M for all x E D. Also, find...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...