Some Examples

After all the explanations about how search is implemented, it is time for some examples.

We start with a search for identities containing the function Cos and its inverse . We restrict the search to elementary functions. Clicking on the following hyperlink will bring up the mathematical search page on the website with the corresponding fields filled in. Search 1

We obtain about 20 results with the first few shown here.

We continue with a search for all integral representations of Euler gamma, where must appear on the left-hand side and a definite integral must appear on the right-hand side. Search 2

We again find about 20 results. Here are some of them.

Next we search for limit representations of the Heaviside theta function (UnitStep in Mathematica). Search 3

Here are the first few of 15 formulas found.

Our next search will be for all formulas with elementary functions that contain the integers 1 to 9. Search 4

About 160 formulas are found. They are mainly products and integrals. Here are the first few.

To find the functional equation of the Riemann zeta function, we search for all identities that contain the Riemann zeta on both sides of the equation. Search 5

We find 13 identities and show the first few. The first establishes a general symmetry, the second that Zeta has its own asymptotics as infinity, and the third is the classic functional equation. Further equations contain finite and infinite sums of zeta functions.

The organization of the Wolfram Functions site lets you easily browse through similar identities for one function. Through the search it is easy to find "equivalent" formulas for groups of functions. This search is for all continued fraction expansions of the inverse trigonometric functions. Search 6

Eleven such expansions are found. Here are some of them.

Our next search contains hypergeometric functions. We will find all identities that contain mathematically. Search 7

Because (as discussed earlier) many thousand potential realizations must be tested, this search will take a few seconds. About 25 matches are found and here are three of them. The formulas returned have arguments that are consistent with those given in the original hypergeometric function.

Our final direct search will be for derivatives of the Bessel function . This means that we want the function and differentiation on the left-hand side of an identity. Search 8

One of the first matches returned is the following formula for the second derivative of .

Clicking the Search for similar formulas button results in about 30 matches. The similarity of the formulas returned consists in either equivalent formulas for the other three Bessel functions, first-order derivatives, or integrals containing simple Bessel functions. Here are the first seven similar formulas. Search 9