www.functions.wolfram.com

Oleg Marichev, Michael Trott

Currently the website www.functions.wolfram.com includes more than 33000 formulas for the about 250 elementary and special functions that are available in Mathematica. 
The structure and organization of the website is discussed in detail. On overview is given how to manage and update this large mathematical knowledge base and how to deliver various formats types (Mathematica notebooks, html, and pdf).
Various "semiautomatic" ways to find new formulas using Mathematica will be demonstrated.
Future development directions will be outlined.

{{"BesselAiryStruveFunctions", 2186}, 
 {"ComplexComponents", 447}, 
 {"Constants", 586}, 
 {"ElementaryFunctions", 9231}, 
 {"EllipticFunctions", 4045}, 
 {"EllipticIntegrals", 687}, 
 {"GammaBetaErf", 3822}, 
 {"GeneralizedFunctions", 162}, 
 {"HypergeometricFunctions", 11098}, 
 {"IntegerFunctions", 1334}, 
 {"MathieuFunctions", 139}, 
 {"NumberTheoryFunctions", 782}, 
 {"Polynomials", 1657}, 
 {"ZetaFunctionsandPolylogarithms", 994}};

Citation

                               Each formula on this web site is given a unique numeric identifier. As this site expands and
                               more information is added, the position of a given formula within its group may change.
                               However, the identifier itself will remain the same, and we recommend that you use the
                               identifier for citation purposes. 

                               To find the identifier for a formula, click on the formula itself. A separate window will appear,
                               giving the formula in Mathematica InputForm and StandardForm notation; the window will also
                               display the formula's unique identifier. The identifier is a combination of digits and dots that
                               describe the position of a given formula in the site hierarchy. For example, "aa.bb.cc.xxxxx.x"
                               identifies the formula cos(z)==sin(Pi/2-z). In this case, aa means that the primary
                               classification is Elementary; bb points to the function Cos[z]; cc points to the section
                               Relations with Other Functions; with Related Functions; with Sin; and the rest of the
                               identifier is a unique serial number. 

                               To see the formula associated with a given identifier, enter its identifier into the search form
                               that you'll find on the top page of this site. 

                               Please cite the formula as in this example: 
                               functions.wolfram.com/01.02.03.12345.1 

                               For electronic documents, you may use the URL form of the identifier to create a live link to the
                               formula itself. 

                               URL form:
                               http://functions.wolfram.com/01.02.03.12345.1 

                               Entering the URL form of a function identifier in an HTML browser will bring you to a window
                               displaying the formula in Mathematica InputForm and StandardForm notations.

Be a contributor
                                                             

                               We would like to see this site become the best resource of its kind on the web. The more
                               comprehensive the site becomes, the more valuable a resource it will be to the mathematics
                               and science community. Many additional features are planned, and a considerable amount of
                               new information will be incorporated in the near future. However, due to the enormous scope
                               of this project, there will always be room for additions and extensions. 

                               If you have results that you think would be appropriate for this site, please consider becoming
                               a contributor. We are especially interested in new results that are specific, concise, and
                               verifiable. Submissions will be reviewed for correctness and are preferred in the form of
                               Mathematica notebooks; however, plain ASCII, TeX, and LaTeX are also acceptable. 

                               We also encourage you to submit interesting examples of how these functions have been put
                               to practical use. Citations to scientific works incorporating or relying on these functions will
                               add historical perspective to their value. 

                               Below are some areas for which contributions would be particularly appreciated, but we
                               encourage you to share your knowledge in any area related to this web site. 














                               Please send your results to functions@wolfram.com. Your help in making this site more useful
                               is appreciated.

0) Historical information about mathematical functions 

                               We are looking for more complete historical information about mathematical functions. For each
                                   , who introduced  or its notation, and when? Who made essential progress in its
                               development---by finding basic formulas, extending its domain of definition, defining key
                               conventions, and so on.... A particular example: who came up with the notation  for the
                               double factorial, and when? 





                               9) More complete tables of particular cases of generalized hypergeometric functions 

                               We are creating the most complete tables possible of particular cases of the generalized
                               hypergeometric functions  Any additional examples would be
                               greatly appreciated. 





                               8) Differential equations for the theta functions  

                               We are looking for ordinary, nonlinear differential equations for the elliptic theta functions 
                                      with respect to  for fixed . 





                               7) Generalizations of Euler and Bernoulli polynomials, Stirling numbers, and partitions
                               numbers for noninteger argument 

                               How is it possible to define the values of Euler and Bernoulli polynomials  and ,
                               Stirling numbers  and ,  and partitions  and  for ? 





                               6) Continuation of derivatives of the psi-function  by order 

                               How is it possible to extend the definition of the derivatives of the psi-function 



                               from integer parameters  to arbitrary complex values ? Such an extension might, for
                               instance, be provided by the formula 



                               where 



                               How do we represent the coefficient  as an analytical function of  in a natural way? 





                               5) Values of derivatives of the zeta function  

                               We are looking for closed-form expressions for the derivatives of the zeta function  of
                               arbitrary order  for integer values of argument :   . 





                               4) Formulas for Mathieu functions and their characteristics  and  

                               We are looking for closed formulas for the Mathieu functions and their characteristics  and 
                                . 





                               3) The inverse function for  

                               The equation  has the solution . Is there a good way to express the
                               solution of the equation ? 





                               2) Series for the inverse error function 

                               We are searching for general formulas for the series expansion of the inverse error function 
                                       around 0. 







                               1) Series for the gamma function 

                               We are searching for general formulas for the series expansion of the gamma function 
                               near its regular points and poles . These formulas should include the
                               psi-functions and their derivatives. 



                               Question 1: How do we express coefficients  through  and  in closed form? 



                               Question 2: How do we express coefficients  through  and  in closed
                               form?  It may be helpful to take into account that

FAQ

                               Where does the material on this site come from?
                                
                                 Much of the material here is from standard references. However, a great deal of the
                                 information was generated specifically for this site and appears here for the first time.
                                 Many other formulas and images appearing here were generated for the first time for
                                 use on the Mathematical Functions of Mathematica poster. 


                               Have these formulas been verified?
                                
                                 The results presented here have all been verified using 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
                                 arbitrary-precision numerical capabilities, and then verify agreement to a very high
                                 level of precision. 


                               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 which is "generically correct" in  will fail only in a domain
                                 of dimension n - 1 or smaller. 


                               In some places where no formula is available, you write No; in others, you write None.
                               What is the difference between No and None?
                                
                                 On this site, No means that we presently have no result that applies, but it does not
                                 rule out the possibility that such a result might exist. None is more specific; it means
                                 that no possible result exists. For example, the function Sin[z] contains no branch cuts
                                 in the finite complex plane, and therefore the branch-cuts entry for Sin[z] is None. 


                               How do I cite from this site a formula that I have used in a publication?
                                
                                 We have a recommended 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.


                               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,
                                 although 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. 


                               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 screen-resolution images simply by copying them, as long as the
                                 copyright notice remains. If you would like higher-resolution renditions for use in
                                 publications, please contact permissions@wolfram.com. 


                               I'm not familiar with some of the notations used on this site. What does  mean?
                               What does  mean? What does  mean? 
                                
                                 In some cases, we have created new compact notations for mathematical properties
                                 like branch cuts, discontinuity sets, and domains of definition. These notations are
                                 described in a separate file on this site. 


                               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. 


                               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. 


                               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. 


                               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 CD-ROM
                                 to be accessed through the Mathematica Help Browser. Watch for further information. 


                               I have a formula or fact that belongs on this site. How do I have it added?
                                
                                 Send it to comments@functions.wolfram.com. 


                               I've found a typo/error/bug. How can I report it?
                                
                                 Please send it by email to comments@functions.wolfram.com.