Analyzing an Identity

Carrying out a mathematical search is possible because in Mathematica notebooks the typeset formulas are unique representations (modulo unimportant choices) of the mathematical meaning of the encoded identities. As a concrete example, here is the cell corresponding to the functional equation of the Riemann zeta function, identity 10.01.16.0001.

Because notebooks themselves are Mathematica expressions, much the same as Sin[x], we can treat documents programmatically. The formulas, identities, and equations contained in notebooks can be converted from their textual (box) representation to semantically meaningful Mathematica expressions.

Here is the formatted form of the cell.

We interpret it and immediately wrap a Hold or HoldForm around the interpreted form to avoid any auto-evaluation, which might change the form of an identity or potentially take a long time for identities that contain integrals.

In the next example, we generate MathML, input form, gif versions, and the traditional form of this identity.

Now we analyze the identity. It is an equality as opposed to an asymptotic expansion or an inequality.

Its fundamental building blocks are the following functions, numbers, and constants.

The identity contains six different numerical functions.

These are all the nontrivial subexpressions of the identity.

The production-quality analysis would continue by removing any dummy variables of summation or integration and by making the identities independent of variables that are not built in, such as in the last example. Because many Mathematica functions come with a different number of arguments, they must be distinguished when analyzing an identity. (For example, the function Zeta is called with one argument in the case of the Riemann zeta function and with two arguments in the case of the Hurwitz zeta function.) Rational numbers are both kept intact as well as taken apart, so that their numerators and denominators can be considered as integers that appear in an identity. Then mathematically identical forms are created (like the power versus square root forms mentioned earlier). As a result, we have a detailed, multifaceted representation of each identity. In addition, we also store information about the section, subsection, ... where the identities have their natural place. Such information is used in the "Search for similar formulas" (see The Results Returned section).