Question

State cauchy's integral formula ​

Answer 1

Cauchy's Integral Formula states that for an analytic function inside a closed contour, the value of the function at any point can be calculated using a specific contour integral around that point.

Step-by-step explanation:

Cauchy's Integral Formula

Cauchy's integral formula states that if a function f(z) is analytic in a simply connected domain, then for any closed path C in that domain and for any point z0 within the path, we have:

f(z0) = 1/(2πi) ∮C f(z)/(z - z0) dz

This equation expresses the value of the function at any point within C solely in terms of an integral around C. The integral is over a closed contour C, enclosing the point z0, where i is the imaginary unit. This powerful result has far-reaching consequences in complex analysis and is used to compute line integrals, evaluate residues, and derive other important results within the field.

Answer 2

Step-by-step explanation:

The Cauchy integral formula states that the values of a holomorphic function inside a disk are determined by the values of that function on the boundary of the disk. More precisely, suppose f : U → C f: U \to \mathbb{C} f:U→C is holomorphic and γ is a circle contained in U.