• Dr. Jay Prakash Jha

Cauchy's integral formula is used to prove the residue theorem which is a result for meromorphic function and a related result, the argument principle. it is known from Morera's theorem that the uniform limit of holomorphic function is holomorphic. This can also be deduced from Cauchy's integral formula. the formula also holds in the limit and the integrand and hence the integral can be expanded as power series in addition the Cauchy formula for the higher order derivatives show that all these derivatives also converge uniformly. Cauchy's integral formula named after Augustine-Louis Cauchy is a central statement in complex analysis. It express the fact that a holomorphic function defined on a disk is completely determined by it's value on the boundary of the disk and it provides integral formula for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration". Complex differentiation like integration behaves well under uniform limit.

Cauchy's integral formula, meromorphic function, Morera's theorem, holomorphic function.

"Importance of Cauchy’s Integral formula for the derivatives of an Analytic function", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.6, Issue 6, page no.139-144, June-2019, Available :http://www.jetir.org/papers/JETIR1907J23.pdf

"Importance of Cauchy’s Integral formula for the derivatives of an Analytic function", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.6, Issue 6, page no. pp139-144, June-2019, Available at : http://www.jetir.org/papers/JETIR1907J23.pdf