Cauchy integral

Cauchy integral (Интегральная формула Коши) is relation for the holomorphic function that expresses its value at some point through the contour integral that encloses this point. It is assumed, that there all contour and the area inside belong to the range of holomorphism of the function.

For holomorphic function \(F\),

\( \displaystyle F(z)=\frac{1}{2\pi \mathrm i} \oint \frac{F(t)}{t-z} \mathrm d t\)

The Cauchy integral is widely used in mathematics and physics. In particular, the representation of tetration through the Cauchy integral allows the efficient evaluation of tetration ^[1].

Evaluation of derivatives

The Cauchy integral allows the efficient evaluation of derivatives of holomorphic functions and the expansion ot the power series: \( \displaystyle F(z)=\sum_{n=0}^\infty c_n (z\!-\!p)^n \)

where \( \displaystyle c_n = \frac{1}{2\pi\mathrm i} \oint \frac{F(z)}{(z\!-\!p)^{n+1}} \mathrm dz \)

It is assumed, that point \( p \) lies within the contour of integration.

In the special case \( p\!=\!0 \) and circular contour of radius \( r \), the formulas above simplify to \(~ \displaystyle F(z)=\sum_{n=0}^\infty c_n z^n \)

\( \displaystyle c_n=\frac{1}{2\pi\mathrm i} \int_0^{2\pi} \frac{F(r \mathrm e^{\mathrm i \varphi})} {(r \mathrm e^{\mathrm i \varphi})^{n+1}} r\, \mathrm i \, \mathrm e^{\mathrm i \varphi}\, \mathrm d \varphi \) \( \displaystyle= \frac{1}{2\pi r^n} \int_0^{2\pi} F(r \mathrm e^{\mathrm i \varphi}) \, \mathrm e^{-\mathrm i \varphi n} \mathrm d \varphi \)