Question

State Cauchy's Integral Formula for Derivatives and use it, if applicable, to evaluate the following integrals,(A) and (B) where C is positively oriented circle of radius 3 centred at 1.

A. ∫ 4iz³e³／(z-i)³∙dz
B. ∫ sin(z)／(z+3i)⁵∙dz

Answer 1

Cauchy's Integral Formula for Derivatives allows us to evaluate certain integrals using contour integration. By applying this formula, we can determine the values of the given integrals (A) and (B).

Step-by-step explanation:

Cauchy's Integral Formula for Derivatives: Let f(z) be analytic inside and on a simple closed curve C, and let z0 be any point inside C. If f(z) is nd-times differentiable at z0, then the nth derivative of f(z) at z0 is given by:

f^n(z0) = (n! / 2πi) ∫(C) f(z) / (z - z0)^(n+1) dz,

where ∫(C) denotes the contour integral around the closed curve C.

To evaluate the given integrals (A) and (B), we can apply Cauchy's Integral Formula for Derivatives:

(A) ∫[C] 4iz³e³/(z-i)³ dz = 12πi * f''(i),

(B) ∫[C] sin(z)/(z+3i)⁵ dz = 2πi * f⁴(-3i),

where f(z) is the function being integrated and [C] denotes the contour of a positively oriented circle of radius 3 centered at 1.