Cauchy's integral formula

This is really a restatement of Cauchy's integral formula.

The ramification index can be calculated explicitly from Cauchy's integral formula.

In general, derivatives of any order can be calculated using Cauchy's integral formula:

Cauchy's integral formula suggests the following definition (purely formal, for now):

According to Cauchy's integral formula, we have:

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.

It generalizes the Cauchy integral theorem and Cauchy's integral formula.

Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions.

Using this powerful tool one may then prove Cauchy's integral formula, and, just like in the classical case, that any vector-valued holomorphic function is analytic.

The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables.

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.

An English title reads as:-"On the extensions of Cauchy's integral formula to analytic functions of several complex variables".

In his following paper, uses this newly defined concept in order to introduce his generalization of Cauchy's integral formula, the now called Cauchy-Pompeiu formula.

The leading term of is obtained by shifting the contour past the double pole at : the leading term is just the residue, by Cauchy's integral formula.

A version of Cauchy's integral formula is the Cauchy-Pompeiu formula, and holds for smooth functions as well, as it is based on Stokes' theorem.

This is significant, because one can then prove Cauchy's integral formula for these functions, and from that one can deduce that these functions are in fact infinitely differentiable.

This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series.

Since the reciprocal of the denominator of the integrand in Cauchy's integral formula can be expanded as a power series in the variable (a z), it follows that holomorphic functions are analytic.

The first version of Montel's theorem is a direct consequence of Marty's Theorem (which states that a family is normal if and only if the spherical derivatives are locally bounded) and Cauchy's integral formula.

In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral.

As the scalar version of Cauchy's integral formula applies to holomorphic f, we anticipate that is also the case for the Banach space case, where there should be a suitable notion of holomorphy for functions taking values in the Banach space L(X).