Mathematica 9 is now available
Outlook
What is implemented?

Polynomials:
     Mathematica in Education and Research 6, n4, 27 (1997)
[Graphics:../Images/TalkGD99_gr_86.gif]:
     Mathematica in Education and Research forthcoming (2000)
Special Functions:
     results partially published in the Wolfram Research Poster Series and notebooks from  
     www.specialfunctions.com and The Mathematica GuideBook (forthcoming)

When is a representation [Graphics:../Images/TalkGD99_gr_87.gif] faithful?

  • Polynomial case: Re or Im
  • General case: more complicated

Implicitly defined functions that can't be solved for [Graphics:../Images/TalkGD99_gr_88.gif].

  • [Graphics:../Images/TalkGD99_gr_89.gif](Lambert/ProductLog function)
  • [Graphics:../Images/TalkGD99_gr_90.gif] (Kepler equation, ε a numeric parameter)
  • [Graphics:../Images/TalkGD99_gr_91.gif] (eigenvalue equation of a square well)
  • [Graphics:../Images/TalkGD99_gr_92.gif] (eigenvalue equation delta function potential)
    ...
  • Solve numerically transcendental equations in a region of [Graphics:../Images/TalkGD99_gr_93.gif]

The special functions of mathematical physics.

  • Again solving odes around branch points
  • Use functional relations
  • Problem for some functions: new branch points at nonprincipal sheets

Pictures of Riemann surfaces of special functions

Other visualization techniques

  • Interactive cuts with projections through 4D [Graphics:../Images/TalkGD99_gr_94.gif]--space
  • Spheres with ears

Pictures of Riemann surfaces over the Riemann sphere

Animations

Animations of Riemann Surfaces with Movable Branch Points

Click on one one of the below equations to see the succession of images that compromise the notebook animation.

[Graphics:Images/RSFAnimations_gr_1.gif]
[Graphics:Images/RSFAnimations_gr_63.gif]
[Graphics:Images/RSFAnimations_gr_155.gif]



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