Properties of Gamma

Complex Analysis · Axiom Academy

LESSON Properties of the Gamma Function One idea organizes them all: is the factorial made continuous — and that single fact fixes its recurrence, its value at , and its poles. 1. The Factorial, Made Continuous Mark the factorials as dots: sitting at x=1,2,3,4,5 . They are isolated — nothing lives between them. The Gamma function is the smooth curve that threads exactly through those dots and fills in every gap, so we can ask for "the factorial of 2.5 " and get an answer. At each integer, Gamma lands on a factorial Defined by Euler's integral for 2. The Recurrence That Builds It The whole curve is held together by one rule: stepping the input up by 1 multiplies the value by the input . That is the continuous echo of , and it works for every x , not just integers — so it lets you climb from any starting value. Step right by 1 → scale the height by x From : each step multiplies by x , regenerating — exactly the factorials. The same step starting at gives , . It drops out of the reflection formula at : . The curve dips to near between , then climbs forever. The reflection formula (where comes from) Set : the right side is , and the left side is . Since there, . 3. Poles Where the Factorial Can't Go Run the recurrence backward : . As the numerator stays near but we divide by , so blows up . Push left again and the blow-ups repeat at every non-positive integer — these are simple poles , and they are exactly the inputs where a factorial never existed.

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