The Gamma Function

Complex Analysis · Axiom Academy

A smooth curve that threads through every factorial — and keeps going where the factorial cannot. What lives between the factorials? You know n! = 1 2 3 n for whole numbers: 3! = 6 , 4! = 24 . But the factorial is only defined on a scatter of isolated dots — the integers. What sits between them? Is there a sensible value for 2 ! , or even (2+3i)! ? Watch the factorial dots appear, then watch a single smooth curve sweep in from the left and thread through every one of them — while also filling in every gap in between. That curve is the Gamma function. To line the integers up, it is shifted by one: it passes through (n) = (n-1)! . The Gamma function is the smooth curve that interpolates the factorial — one continuous graph agreeing with (n-1)! at every positive integer. Walk along the curve — and meet the poles Drag the point left and right along the real axis. The readout shows (x) . Notice it lands exactly on (n-1)! at every integer, and obeys one clean rule everywhere: (x+1) = x\, (x) . Push toward x = 0 and the value rockets to infinity — the first of the poles the curve has at 0, -1, -2, At every positive integer the curve hits a factorial; toward each non-positive integer it blows up — a simple pole at 0, -1, -2, One integral builds the whole thing

This is the written version of the interactive lesson above. See the full Complex Analysis course.