Back to . . . .  NCB Deposit  # 121 Dr. Cye Waldman cye@att.net "On Plane Curves in the Complex Plane" Bessel 1784-1846

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

 Hermite Polynomial 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.

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  < http://curvebank.calstatela.edu/waldman/waldman.htm >
 2012