Leonhard Euler

Gamma function

The birth of the gamma function, by Leonard Euler in the year 1729, was due to the merging of several mathematical streams. The first was that of interpolation theory, a subject focussing on the extension of functions from the domain of natural numbers to the domain of real number. The second stream was that of the integral calculus and of the systematic building up of the formulas of indefinite integration.

The problem was to find a function for the $n-th$ term of the following series: \[ 1,2,6,24,120,720,.... \]

Euler's task was to find an analytic expression, (an expression which could be derived from elementary manipulations with addition, subtraction, multiplication, division, powers, roots, exponentials, logarithms, differentiation, integration, infinite series), which would yield factorials when a positive integer was inserted, but which would still be meaningful for other values of the variable.

In his article On transcendental progressions that is, those whose general terms cannot be given algebraically he gave the following infinite series solution: \[ \frac{1.2^n}{1+n}\frac{2^{1-n}3^n}{2+n}frac{3^{1-n}4^n}{3+n}\cdots \] This is clearly and infinite process, for evaluating $n=3$ we get: \[ \frac{2.2.2}{1.1.4}\frac{3.3.3}{2.2.5}\frac{4.4.4}{3.3.6}\frac{5.5.5}{4.4.7}\cdots=6 \] One clearly sees that all terms are cancelled except the terms of the factorial. However this formula does not help to find easily the $n-th$ term, it only extends the case of a positive integer number, to any positive number.