the gamma duplication formula

The duplication formula relates the gamma value at 2z to two gamma values at z and z + 1/2: Gamma(z) Gamma(z + 1/2) = 2^(1 - 2z) sqrt(pi) Gamma(2z). It is Legendre's identity, and it is the gamma-function shadow of the way the factorial of an even number relates to products of half-integers. Where the functional equation steps the argument by 1, the duplication formula HALVES it — it tells you how Gamma behaves when you split a doubled input into two staggered halves.

Think of it as bookkeeping for products. Group the integers up to 2n into evens and odds: the evens give 2 * 4 * ... = 2^n n!, the odds give the double factorial. Gamma packages both kinds of product, and the duplication formula is what falls out when you demand consistency between Gamma(2z) on one side and the pair Gamma(z), Gamma(z + 1/2) on the other. A clean way to see it is via the Weierstrass product or via the beta integral; either route produces the constant 2^(1-2z) sqrt(pi) that makes both sides agree, and the sqrt(pi) is once again the fingerprint of Gamma(1/2).

The formula generalises. Gauss's multiplication formula does the same trick splitting Gamma(nz) into n staggered gamma values Gamma(z) Gamma(z + 1/n) ... Gamma(z + (n-1)/n), with duplication being the case n = 2. These identities are essential whenever a calculation produces gamma values at half-integer or rational shifts and you want to collapse them — they appear constantly in hypergeometric evaluations, in the volumes of spheres, and in tidying up beta-function answers. A caution: the constant on the right is genuinely 2^(1-2z) sqrt(pi), not a 'nice' integer; getting that power of 2 right is the whole content of the formula.