Cauchy's integral formula is a fundamental result in complex analysis that provides a way to evaluate integrals of analytic functions over closed curves. It states that if a function is analytic inside and on some simple closed curve, the value of the function at any point inside the curve can be expressed as a contour integral of the function around the curve. The formula is often written as f(a) = \frac12\pi i \oint_C \fracf(z)z-a \, dz , where f(z) is the analytic function, C is the closed curve, and a is a point inside C . This powerful tool simplifies the computation of integrals and has numerous applications in mathematics and physics.