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 NCB Deposit  # 126

Dr. Cye Waldman
cye@att.net


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The Compleat Gamma Pulse  

Spirals, Yin-Yang Symbols and Globules


Fermatstamp
The gamma pulse is a model transient pulse that was designed for application to physical systems. It can be convolved with impulsive solutions in temporal or harmonic space to develop the transient response to an input pulse. It has many interesting mathematical properties as well, and can be used as a plane curve generator in the complex plane. In this brief note we discuss application to plane curves and physical pulses. Thus, without any further ado, we introduce the gamma pulse equation.

gammaequation

( Click here for the full "pdf" Article including Equations and Matlab
Code. )
Animations
Yin-Yangsymbol
Yin-Yang Symbol
Yin-Yangreverse
Evolution

of the

Yin-Yang Symbol
GammaGlobules
Gamma Globules

An alternative take on the Yin-Yang curve:
Banakh et al  prove that Fermat's spiral is a unique line in the class of smooth algebraic curves in polar coordinates.

Banakh

spirals

In physical systems, a pulse can be adequately defined with two parameters: the rise of time and the pulse width.

A sampling of pulses may be useful to those seeking applications of the gamma function.
pulsegraph



References, Comments and Matlab Code

Initial Matlab Code for The Compleat Gamma Pulse:

p=inline('tau.^n.*exp(-tau)','tau','n');
tau=linspace(0,10,1e6+1)';
n=10+10i;
figure;plot(p(tau,n));axis equal

The specific Matlab code for the yin-yang is
p=inline('exp(n*log(tau)).*exp(-tau)','tau','n');
tau=logspace(-11,1,1e6+1);
n=i/pi;
pyy=p(tau,n);
q=[pyy;-flipud(pyy)];
figure;plot(q);axis(1.1*[-1 1 -1 1]);axis square

This material was written without knowledge of a parallel article.  However, we now must acknowledge finding the following paper.  After all, Yin-Yang animations are popular.
< Fermat's_Spiral_&_Yin-Yang.pdf >
Banakh, T., Verbitsky, O. and Vorobets, Y., Fermat's Spiral and the Line Between Yin and Yang, arXiv:0902.1556v2, 2009.
Oldham, K. B. and Spanier, J., The Fractional Calculus, Dover, 1974.
Other Spiral curves  < http://curvebank.calstatela.edu/spiral/spiral.htm >
Other Waldman curves and articles:
< http://curvebank.calstatela.edu/waldman/waldman.htm  >
< http://curvebank.calstatela.edu/waldman2/waldman2.htm >

  
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