Cauchy's Integral Formula is a theorem in complex analysis that relates the values of a holomorphic function inside a closed contour to the values of the function on the boundary of the contour. It is named after the French mathematician Augustin-Louis Cauchy.
A holomorphic function is a complex-valued function that is differentiable at every point in its domain. This means that the function is smooth and has no sharp corners or breaks. In other words, it is a function that can be represented as a power series.
Cauchy's Integral Formula is used to calculate the values of a holomorphic function at any point within a closed contour, given the values of the function on the boundary of the contour. It is a powerful tool in complex analysis and has applications in various areas of mathematics and physics.
The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex-valued function to be holomorphic. Cauchy's Integral Formula is a consequence of these equations and can be used to prove the analyticity of a function.
Yes, Cauchy's Integral Formula can be extended to functions with singularities, but the contour of integration must be modified to avoid the singular points. This extension is known as the Cauchy integral theorem and is an important tool in the study of complex analysis.