Prove Cauchy's Integral Theorem for k-connected Jordan domains: Let I be a k-connected Jordan domain and...
4. (a) Use Cauchy's residue Theorem to provide alternative proofs for Cauchy's integral formula and its extension. (b) Evaluate z+ 4z + 5 dz, son z2 + z where C is the positively oriented circle of radius 2 centered at the origin.
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...
Evaluate the integral. Does Cauchy's theorem apply? Show details . 2 & de 1 6 z dz > ¿ z2+ CZ til: i Z2+1 C: 12-11 Counterclockwile Counter clock wise
Problem A.5. Let D be a region in the complex plane. (a) State Green's theorem in terms of f(2)u(, y) + iv(x, y),z-+ iy, and (b) Prove the following case of Morera's theorem: If f is continuously differentiable 0 for every circle γ in D, then f is analytic in D. Hint: in D and J,f(z)dz Use part (a).
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
please 2 only, thanks Exercises dA (1) Use Cauchy's residue theorem to compute Jo 2+sin (2) Repeat the preceding exercise for 8" 131. (3) Let a be a complex number such that lal < 1. Prove that (2 27 Jo 1 - 2a cos 0 + a2d6 = 1 - 22 (4) What is the value of the integral in the preceding exercise when |al > 1? (Hint: Let b= 1.)
8. Let A be an integral domain containing elements x, y, and z. Prove the following facts. (a) If z|x and zly, then x/2 + y/2 = (x + y)/2. (b) If 2 x, then y. (x/2) = (y • x)/2. (c) If yız and x[(z/y), then (x • Y)|z, and 2/(x • y) = (z/y)/x.
Question of 9 Laurent Series and the Residue Theorem - 9.4 Argument Principle. I want #2 to be answered. Exercises 9.59. 1. If f(2) is analytic inside and on the simple closed contour C, and f(z) on C, show that the number of times f(z) C is given by assumes the value a inside f'(2)dz. 1 2πί Jσ f(2)- simple closed contour C except for finite number of poles inside C. Denote the zeros by z1,. . , Zn (none...
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...
3-2. Prove Theorem 3.2. Theorem 3.2 Let I S R be an open interval, xe I, and let f. 8:1\{x} → R be functions. If there is a number 8 > 0 so that f and g are equal on the subset 12 € 7\(x): 13-X1 < 8 of I\(x), then f converges at x iff g converges at x and in this case the equality lim f(x) = lim g(z) holds.