On Plane Curves in the Complex Plane

Bessel
1784-1846

Click here for the full article including Equations.

Bessel Function Curve

"Without apology or embarrassment, he (Euler) treated these numbers (real and imaginary) as equal players upon the mathematical stage and showed how to take their roots, logs, sines, and cosines."		"In mathematics you don't understand things. You just get used to them."
W. Dunham in Euler: The Master of Us All		Johann von Neumann

Additional examples:

Hermite Polynomial
Hermite equations

The analytical signal of the Hermite polynomial for n = 50 looks
remarkably like a plane curve obtained from the Cauchy pulse with n = 100.

A Three-Dimensional Hermite Polynomial

There are many paths to complex functions with fractional calculus.

References, Comments and Matlab Code

Matlab Code for "The Apple of My i" can be expressed in only four lines. We will repeat it here for your convenience.

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

Please see the Bibliography in the accompanying PDF file for a complete set of references.

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

Dunham, William, Euler: The Master of Us All, Mathematical Association of America, 1999.

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 NCB Waldman animated curves < ..//waldman/waldman.htm >

2012