Cauchy Method

Cauchy Method of evaluation of superfunction refers to the real–holomorphic transfer function $T$ that has no real fixed point, but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches $L$ at $\mathrm i \infty$ and approaches $L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

Description

The transfer equation for the superfunction $F$ of transfer function $T$ Can be written as follows:

$F(z+1)=T(F(z))$

Within some domain of values of $z$, the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z))$

Then, for some strip in direction of imaginary axis, the transfer equation can be written through the integral Cauchi

$\displaystyle F(z)=\oint \frac{F(t)}{t-z} \mathrm d t$

where the integration is performed along the boundary of the strip. In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$; for long "strip" this is supposed to give the precise approximation of the integral. The width of the "strip" along the real part of $t$ should be equal to 2; then, the transfer equation and its inverse allow to represent values a the edges of the "strip" through the values along the line of the strip. In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation; that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.