information:highlighting, quality:correction
Motivation

The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve y = f (x) {\displaystyle y=f(x)} that connects the points of the factorial sequence: (x, y) = (n, n !) {\displaystyle (x,y)=(n,n!)} for all positive integer values of n {\displaystyle n}. The simple formula for the factorial, x ! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function f (x) = Γ (x + 1) {\displaystyle f(x)=\Gamma (x+1)}.

The gamma function is not only smooth but analytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin (m π x) {\displaystyle k\sin(m\pi x)} for an integer m {\displaystyle m}. Such a function is known as a pseudogamma function, the most famous being the Hadamard function.

A more restrictive requirement is the functional equation which interpolates the shifted factorial f (n) = (n − 1) ! {\displaystyle f(n)=(n{-}1)!} : f (x + 1) = x f (x) for any x > 0, f (1) = 1. {\displaystyle f(x+1)=xf(x)\ {\text{ for any }}x>0,\qquad f(1)=1.}

But this still does not give a unique solution, since it allows for multiplication by any periodic function g (x) {\displaystyle g(x)} with g (x) = g (x + 1) {\displaystyle g(x)=g(x+1)} and g (0) = 1 {\displaystyle g(0)=1}, such as g (x) = e k sin (m π x) {\displaystyle g(x)=e^{k\sin(m\pi x)}}. One way to resolve the ambiguity is the Bohr–Mollerup theorem, which shows that f (x) = Γ (x) {\displaystyle f(x)=\Gamma (x)} is the unique interpolating function over the positive reals which is logarithmically convex (super-convex), meaning that y = log f (x) {\displaystyle y=\log f(x)} is convex (where log {\displaystyle \log } is the natural logarithm).