Cauchyformeln

Cauchyformeln, also known as Cauchy's integral formulas, are fundamental results in complex analysis named after Augustin-Louis Cauchy. These formulas provide a means to evaluate complex integrals and are central to the theory of holomorphic functions.

The principal Cauchy integral formula states that if a function \(f\) is holomorphic inside and on a simple closed contour \(C\), and \(a\) is a point inside \(C\), then the value of \(f\) at \(a\) can be expressed as:

\[

f(a) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - a} \, dz

\]

This formula not only allows the computation of function values but also leads to various derivatives’ formulas. The n-th derivative of \(f\) at \(a\) can be obtained using:

\[

f^{(n)}(a) = \frac{n!}{2\pi i} \int_C \frac{f(z)}{(z - a)^{n+1}} \, dz

\]

Cauchy’s integral formulas are vital for establishing the properties of holomorphic functions, including their analytic continuation and the maximum modulus principle. They also serve as a foundation for various other theorems in complex analysis, such as Liouville's theorem and Morera's theorem.

In summary, Cauchyformeln provide a powerful tool for evaluating and understanding functions of a complex variable, emphasizing the interconnectedness of function values and their integrals around closed contours.