


Q:  Have these
formulas been
verified? 

The results presented here have all been verified with
Mathematica
to be generically correct. In many cases it is possible to simply evaluate an equation in
Mathematica, either by entering it directly or by applying the command
FullSimplify.
In[1]:= Sin[2x] == 2Cos[x]Sin[x]
Out[1]= True
When it is not possible to verify the formulas directly this way, we have
verified them analytically, using our knowledge of the properties of the functions involved
to select test points intelligently, evaluate the expressions using Mathematica's
arbitraryprecision numerical capabilities, and then verify agreement to a very high
level of precision.

Q:  What
does it mean to say that the
results here are "generically correct"? 

For example, Mathematica evaluates the expression x/x == 1 to be
True. However, the equation fails when x = 0 since the value of 0/0
is indeterminate. Since the equation is valid over the finite complex plane but invalid
at a single point, the result is said to be generically correct.
In general, an equation that is "generically correct" in
will fail only in a domain of dimension
n  1 or smaller.

Q:  How do I cite from this site a formula
that I have used in a publication? 

We have recommended a citation
format. Using this format
will ensure that the citation remains valid through time, even if the position of a given
formula within its section or subsection changes as new formulas are added.

Q:  What do you mean by InputForm
and StandardForm? 

InputForm and StandardForm are two
standardized notations used by the Mathematica system. Formulas are presented
here in these two forms.
InputForm expressions contain only the characters available on a standard
ASCII keyboard. Expressions in InputForm can be typed directly into the
Mathematica system without the use of any special key sequences.
StandardForm notation is a refinement of traditional notation
with certain ambiguities removed. Expressions in StandardForm can
use special mathematical characters and positioning. StandardForm
is the default notation used in Mathematica notebooks. However,
entering
some expressions in StandardForm requires the use of special keyboard commands for positioning subexpressions or entering special symbols.
Here is an expression in InputForm.
Integrate[1/x^2, {t, 0, 1}]
Here is the same expression in StandardForm.
For more information about notation in Mathematica, see
Section
1.10.9,
"Forms of Input and Output," in The Mathematica Book.

Q:  Why can I only download the graphics
Mathematica code that generates
the graphics, instead of the graphics themselves? 

An evaluated notebook that contains all visualizations for a single function
has an average size of
about 300 MB (the size of the notebooks of some functions approaches
up to 1 GB). Even when all the
Postscript graphics are replaced by bitmap graphics, the average size
of the notebooks is about 15
MB.
Downloading such large documents is not practical. But
each of the visualization
notebooks can be easily evaluated and will produce all the graphics shown on
the web pages. Mathematica can also export these graphics in various formats.

Q:  I
would like to use some of the graphic
images on this site for my teaching.
How do I get permission? 

You may use the screenresolution images simply by copying them, as long as the
copyright notice remains. If you would like higherresolution renditions for use in
publications, please contact
permissions@wolfram.com.

Q:  This site would be more effective if it
were presented using MathML, an extension to
HTML for describing mathematical expressions. Do you plan on including MathML in
future versions of this site? 

Yes, we do, as soon as there is widespread browser support for rendering MathML
expressions. Mathematica's command HTMLSave
already allows for
the automatic generation
of MathML code for mathematical expressions. The technological barrier at present
is the lack of MathML support in the major browsers.

Q:  What is the best way to print a hard
copy of results from this site? 

At the top of the page for each function there is a link that will let you download from
within your web browser the complete contents of the page in PDF format. Viewing
a PDF file requires Adobe Acrobat Reader.
Printing from within Acrobat Reader will
give you output to the full resolution of your printer; printing from within your
browser will give you output only at screen resolution.

Q:  Can I mirror
this site onto a local
server for my university or research institution? 

No, you can't. Wolfram Research is hosting this free site as a service to the mathematics
and science community but retains the rights to all files within it.

Q:  This site was generated from
Mathematica notebooks. How can I get a copy of
these source notebooks for my own use? 

The notebooks from which this site is generated include additional data needed to construct
and update the site, and we will not release these source notebooks in their raw form.
However, we plan on producing a version of these notebooks on CDROM to be accessed
through the Mathematica Help Browser. Watch for further information.




© 1998 Wolfram Research, Inc.
