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.

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.