PLS show your work 4. Let / be an entire function on C such that 11...
Please be neat and show all work. I am trying to understand this material. 8. Use Definition 2 to prove that limz1+i (6z - 4) 2+6i. Definition 2. Let f be a function defined in some neighborhood of zo, with the possible exception of the point zo itself. We say that the limit of f(z) approaches zo is the number wo and write 20 lim f(z) = wo or, equivalently, as 02 2 0n(2)f if for any & > 0...
7. Let f be an entire function. Suppose there exists € >0 such that f(2) > € for every 2 E C. Show that f is constant. (Hint: Apply Liouville's theorem to the function g(2) = 1/f().)
d o ille Tunctions on to the effect that every bounded harmonic function on the whole complex plane has to be a constant function. (11) (Fundamental Theorem of Algebra) Let P be a polynomial given by de the unit circle P(x) = do +212 + a2z2 + ... + anz", ZEC, where do, 21, 22, ...,an, are complex constants and n is a positive in- teger. Prove that P has a zero in C in the sense that there exists...
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...
2. Let U C R2 be simply connected and let to E U. Let g: U(oR2 be irrotational and of class C1. Assume that there exists r >0 such that B(zo, r) C U and g=0. Let γ be a closed sinile polygonal arc with range in U \ {zo), let「be its range, and let V be the bounded connected component of R2 \ Г. (a) Assume that V C U \ [xo) and prove that g=0. (b) Assume that...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
complex analysis, cite all theorems used Let fcz) be an entire function and there exists a real number Ro such that Ifcail sizl for any complex number z 12/7RO Prove that f is of the form VZEC with f(Z)= arbe
1 Let A-I: 11 (a) Is the vectoran eigenvector of A? Show your work 2 3] 2 (b) Is the number 4 an eigenvalue for A? Show your work.