Cauchy Method - Revision history

Kouznetsov: /* Description */

2016-02-06T13:11:14Z

Description

Kouznetsov: /* Description */

2016-02-06T13:10:59Z

Description

Kouznetsov: /* Description */

2016-02-06T13:10:38Z

Description

Kouznetsov: /* Description */

2016-02-06T13:10:17Z

Description

Kouznetsov at 13:09, 6 February 2016

2016-02-06T13:09:50Z

Kouznetsov at 13:09, 6 February 2016

2016-02-06T13:09:07Z

Kouznetsov: Load stub

2016-02-06T13:08:34Z

Load stub

New page

[[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
$L^*$ 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]]

$
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.

==References==
<references/>

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html 
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf 
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf 
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

@@ Line 27: / Line 27: @@
 the transfer equation can be written through the [[integral Cauchi]]
+$\displaystyle
 F(z)=\oint \frac{F(t)}{t-z} \mathrm d t
+$

@@ Line 30: / Line 30: @@
 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 $L^*$;

@@ Line 22: / Line 22: @@
 the "inverse" equation takes place,
-$F(z-1)=T^{-1}(F(z)$
+$F(z-1)=T^{-1}(F(z))$
 Then, for some strip in direction of imaginary axis,

@@ Line 3: / Line 3: @@
 The method is supposed to evaluate superfunction that approaches
 $L$ at $\mathrm i \infty$
-and
+and approaches
 $L^*$ at $-\mathrm i \infty$

@@ Line 1: / Line 1: @@
-[[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 $L*$.
+[[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
@@ Line 6: / Line 6: @@
 $L^*$ at $-\mathrm i \infty$
-Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/mathrm e)$.
+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.