The Apple of My i

Cauchy
1789-1857

Waldman investigates spirals graphed in the imaginary plane .

(Click here for the full Article including Equations and Matlab Code)

"The Apple of My i "

Additional examples of curves generated by the Cauchy pulse are shown below.

Lattice π

We obtain a curve somewhat like the limaçon if we conside the π th derivative of the Cauchy pulse (n = π).

The image appears to consist of seven overlapping leaves though there are very few in nature.

The Centurion
If we allow real umber n-values to increase, we obtain increasing number of nested loops. This animation is for n = 100.

Centurion

References, Comments and Matlab Code

Matlab Code for "The Apple of My i" can be expressed in only four lines:

psi=inline('1./(1-i*tau).^(n+1)','tau','n');
tau=linspace(-1000,1000,1e6+1)';
n=i;
figure; plot(psi(tau,n)); axis equal

Or, in terms of the angular variable:

psi=inline('cos(theta).^(n+1).*exp(-i*(n+1)*theta)','theta','n');
theta=linspace(-pi/2,pi/2,1e6+1)';
n=i;
figure; plot(psi(theta,n)); axis equal

Hahn, Stefan, Hilbert Transforms in Signal Processing, Artech House, 1996.

Oliver, Frank, et al, NIST Handbook of Mathematical Functions, Cambridge, 2010.

Weisstein, Eric , CRC Encyclopedia of Mathematics, CRC Press, Chapman and Hall, 2003.

Zwikker, C., The Advanced Geometry of Plane Curves and Their Applications, Dover Press, 1963.

Other Sinusoidal curves < ..//spiral/spiral.htm >

2012