Math Games

The Wolfram Functions Site

For the sine function, most people think about a specific image.

Figure 1. The Sine function

For the expanded Wolfram Functions Site (functions.wolfram.com), Michael Trott offers many more visualizations.  One of Michael's experiments: show the behavior of Taylor Series as more terms are added.  The lower part of the picture uses relatively few terms -- at the top, many terms. For sine, spirals approach specific points. The experiment makes a variety of interesting behaviors for each function, but hasn't been studied in depth. More than 10,000 other function visualizations are available, and many more are on the way. Complete code is offered for every visualization. 

Figure 2. Taylor Series accuracy of the Sine Function, as described at Sin/visualizations/14.html.

Many of these visualizations are new to me.  I had seen Weyl Sums, but didn't know the name. Saunders Graphics were completely new to me.  Integral curves of Newton's second law for ProductLog gives an unexpected MasterCardiform. Contour plots of Padé approximant arguments are gorgeous. Michael's Riemann Surfaces have won awards.  All of these visualizations can add understanding to properties of a given function.

Figure 3. Saunders Graphic of ArcSin.

I have a tape of Mandlebrot Zooms that I often lend to parents of young children. A typical comment, about a six year old.  "She called one of them the dancing elephants, and watched it several times." A good influence, I think.  Of course, the functions site has Mandlebrot-type pictures of each function.

Figure 4. The Mandlebrot visualization of Hyberbolic Sine.

The functions site isn't just pictures -- almost 90,000 functional identities are given, including many varieties of Integral forms.  Oleg Marichev, author of a classic 5-volume integral encyclopedia, has added many Integral forms to the functions site.  His comprehensive study of Integrals goes well beyond what could be packed in 50 volumes -- vastly more than the Handbook of Integer Functions by Abramowitz and Stegun.  With the aid of specialized Mathematica programs, Oleg discovered thousands of new functional identities, many of which have now been published for the first time.  A handful are classic results, now completely correct for the first time.

Figure 5. Evolute of Sine.

Algebra and trigonometry provide tools to solve elementary mathematical problems.  For more difficult problems, more powerful tools are needed.  In the world of mathematics, "special functions" are those functions that show up over and over again, or which are particularly powerful.  In the SIAM $100 challenge, out of ten seemingly impossible math problems, seven of them were directly solved by well-known special functions. Here are a few of the special functions covered in depth at the functions site.

• Elliptic integrals arise from simple electrostatic and magnetostatic problems
• Legendre functions arise in the description of a penny rolling on a table
• Hermite polynomials arise in the quantum mechanical version of the harmonic oscillator
• Chebyshev polynomials arise frequently in numerical analysis
• Trigonometric functions arise in the description of square-shaped drums
• Bessel functions arise in the description of circular drums
  
• Mathieu functions arise in the description of ellipse-shaped drums
• Polylogarithmic functions arise in the descriptions of elementary particles
• Meijer G functions are the top echelon generalized function -- they can substitute for almost any other function.
  

What else... it's the largest existing MathML site.  It's all free (though citations are requested).  As functional identities, the new discoveries can be used in any language, even COBOL or Jovial. The site is growing. It is crosslinked with Mathworld. PDF files and Mathematica notebooks are offered to summarize every function. 

For more, you can see the news release, the history, the overview, or the people behind it. Or the site itself.

References:

Mathematica Code:

Complete code (by Michael Trott) for all visualizations is available within each Visualization section.

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Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com. He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.