Suppose that ƒ and g are analytic in the region A and suppose that g'(z) ± 0 for all z Є A; Suppose that g is 1-1 and let y be a closed curve in A. Prove that for z 1. f(z) · I(v; z) = g'(z) Σπί ƒ(S) ds. Hint: Use the Cauchy Integral Formula and apply it to h(C) = ƒ(C)(S-2) 9(S)-9(2) forz = S and h(5) = f(S). Here I (Y; zo) g'(5) I(Y; zo) times. Use this result and apply it to the case where g(z) = e². 2πi Sy dz 2-20 " and winds around zo,

Q: y, z) = x*z and let C be the curve given by r(1) = (2e', 2V21, ) för 0 <i< 3. Côm bolic notation and…

A: In the given question, the concept of line integral is applied, Line Integral The function to be…

Q: Suppose the function y = h(x) is nonnegative and continuous on (a.B), which implies that the area…

A:

Q: Indicate whether the following statement is true or false. Let f (x, y) be any real valued function…

A:

Q: (3m/2.0,0)4 (m/2. m/2,0) (0.3=/2.0) 2. Let F(x, y, z) = (3x²y - z sin(x))i + x³ + cos(x)k and let C…

A:

Q: Let f(x, y, z) = x²z and let C be the curve given by r(t) = (2e', 2√2t, 2e) for 0 ≤ t ≤ 3. Compute…

A: , The curve C is given by  for .To find:.

Q: f(y) - f(z) = [²√ 1 (² + 1 dt = So f(z+t(y-2)) dt = ₁ Df (z+t(y-2)) dt (y-2).

A:

Q: 44. Let 8(t) be a continuous function such that 4-y² S.J 8(t) dt = 57. Use polar coordinates to…

A:

Q: Compute the length of the curve r(t) = = (2t, In(t), t²) over the interval 1 ≤ t ≤ 8. (Use symbolic…

A:

Q: Given gamma is the curve defined by R(t) = ti + tj + tk, for t [1, infinity). How would I evaluate…

A:

Q: 3. Let f : R→ R be continuous and let a > 0. Define g : R → R by rx+a g(x) = f(t)dt, x € R. x-a Show…

A:

Q: Consider the one-form w = dx +g' (y) dy on R², where g : R →Ris an arbitrary smooth function. Show…

A:

Q: 1. If F(x, y) = (3+2xy)i + (x² – 3y²)j. Evaluate F. dr, where C is the curve given by r(t) = e'…

A:

Q: Let C: [o,2]> R be a curve defined by C(t) = (t²,t'). Evaluate. Sy dx x dy.

A: Given:  C: [0, 2]→ℝ2   is a curve defined by                C(t)=(t2, t3)implies x=t2 and…