Chapter 10

Hitting the Ends

Folkmar Bornemann

Problem 10

10.1 A First Look: Why Not Monte Carlo? · 10.2 Making It Deterministic · 10.3 Solving It Numerically · 10.4 Solving It Analytically I: Separation of Variables · 10.5 Solving It Analytically II: Conformal Mapping · 10.6 The Joy of Elliptic Functions · 10.7 Finale à la Ramanujan: Singular Moduli · 10.8 Harder Problems

Unexpectedly, there is a closed-form solution:    .
The first 10,002 digits.

Code

• C
   
  • Random walk on spheres (used to generate Table 10.1)
     
• Maple
   
  • Session: p. 215
     
• Mathematica
   
  • Sessions: pp. 215, 216
  • Table: 10.4 (check of)
     
• MATLAB
   
  • Sessions: pp. 205/206, 208, 222
  • Random walk on spheres (as alternative to C)
  • Solution using the Schwarz-Christoffel Toolbox
  • Tables: 10.2, 10.3
  • Figures: 10.1, 10.2, 10.5 (left), 10.5 (right)
  • Commands: richardson, poisson, cot_formula

Movie

A sample path (Figure 10.1 in action)

Links

• Toby Driscoll: Schwarz-Christoffel toolbox for MATLAB.

• The Modular Elliptic Function λ. From functions.wolfram.com―A Wolfram Web Resource.
   
• Eric W. Weisstein. Elliptic Lambda Function. From MathWorld―A Wolfram Web Resource.
   
• Mark B. Villarino. Ramanujan's Most Singular Modulus. arXiv:math.HO/0308028. (A detailed, elementary, and self-contained account on Ramanujan's famous formula for k₂₁₀.)

Historic Sources

• Augustin-Louis Cauchy: Sur quelques propositions fondamentales du calcul des résidus, Exerc. Math. 2 (1827), 245–276; Oevres (2)7, 291–323 (1889). Formulas (86/87) and (110).
   
• Carl Gustav Jacob Jacobi: Fundamenta nova theoriae functionum ellipticarum, Bornträger, Regiomontum, 1829. p. 108.
   
• Heinrich Weber: Elliptische Functionen und algebraische Zahlen, Vieweg, Braunschweig, 1891. pp. 149 and 502.

Supplements

• Footnote 80: Joseph Hersch's geometric determination of p = 1/6 for a √3 x 1 rectangle (using conformal transplantations)
   
• Folkmar Bornemann. An Experimental Approach to the Singular Modulus k₁₀₀. Note of March 1, 2005.
   
• Encounters of a Numerical Analyst — From Monte Carlo to Ramanujan. Lecture on topics of Chapter 10.

Errata

• What Figure 10.1 should have looked like.

This is a web page for the book:

Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jörg Waldvogel: The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing. Society of Industrial and Applied Mathematics (SIAM), Philadelphia, 2004.

last modified: 04/23/07 14:41 +0200 (FB)