SIN (20 pts) Use the principle of deformation of path and Cauchy's Formula to evaluate the...
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.
Question 4.1.2 please.
4.1 Use Cauchy's intergral formula (or its extensions) to compute the following integrals (each contour is positively oriented) 4.1.1 2z -dz 4.1.2 Katie1 (7+*-dz
Prove Cauchy's Integral Theorem for k-connected Jordan domains: Let I be a k-connected Jordan domain and f(2) be analytic in some domain containing 12. Then, Son f(z)dz = 0. Hint: Use the Deformation Principle.
2. More integrals! Evaluate each integral, using either a Cauchy Integral Formula or Cauchy's Residue Theorem. Take C to be the circle [2] = 3, oriented counter-clockwise. 1) Sota-1jad: 6) Se TH h) Sorºcos(1/2)da
The circuit C, depicted below, is traversed clockwise starting and ending at -1. It consist of two parts: C = A + B where A c {z = z + iy : y = 1-2 and (a) Give parametrisations of A and B (b) Evaluate the line integrals L,-/ Re(z) dz, L,-, Re(z) dz. 1+2 (1+z+ z2)2 Calculus for complex line integrals to evaluate (c) Let f be given by f(z) - 2% Use the Fundamental Theorem of (d) Does...
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?
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2 & de 1 6 z dz > ¿ z2+ CZ til: i Z2+1 C: 12-11 Counterclockwile Counter clock wise
(2) Apply Cauchy's Integration to EVALUATE the following INTEGRAL: 27 1 dt. 0 3 – sin(t)
Use the theory of residues to evaluate the following real, definite integrals: 210 do Hint: integrate on the unit circle, z = elo; dz = izdo. Hint: find the residue in the upper half plane. ſº I dx 13+5 sin e dx b. So y2 + 4x +5 J-ox? +dx
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NOT use a calculator. Exact answers only, no decimals.
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr