A powerful way to prove a mathematical result (e.g. an identity of the form A=B) is to introduce a new object or concept (say C) and connect it in two different ways to the original problem. For instance, if one can show that A=C and one can also show that C=B, then one can deduce that A=B. More generally, one can introduce n new objects or concepts, and establish at least n+1 non-trivial connections between these objects and each other, or to the original problem; for instance, if one introduces two new objects C,D and three connections, two of which A=C, D=B are to the original problem, and one of which C=D is between the newly introduced objects, then one has again established A=B.

A typical example of this is the use of complex analysis methods to solve a real analysis problem, as per Hadamard’s famous dictum “The shortest path between two truths in the real domain passes through the complex domain”. For instance, suppose one wants to compute some real integral A and show that it equals some value B. To do this, one can introduce two new concepts (the complex contour integral C, and the notion of a residue of a pole D); write the real integral as a contour integral (establishing a result of the form A=C), invoke the residue theorem (which is a result of the form C=D), and then compute the residues (a result of the form D=B), to obtain the final desired result A=B.

— Terence Tao

2013.05.11 Saturday ACHK