the gamma reflection formula

The reflection formula is a startlingly clean bridge between the gamma function and ordinary trigonometry: Gamma(z) Gamma(1-z) = pi / sin(pi z), for every non-integer z. On the left, two gamma values at inputs that add up to 1 — z and 1-z are mirror images across the point 1/2. On the right, the elementary sine function. It says the product of those two transcendental gamma values is nothing more exotic than pi divided by a sine. This is one of the most beautiful identities in all of analysis.

Why is it true, in spirit? The reciprocal 1/Gamma is an entire function whose zeros sit exactly at the integers 0, 1, 2, ... and 0, -1, -2, .... The product 1/(Gamma(z) Gamma(1-z)) therefore has zeros at every integer, positive and negative — exactly the zero set of sin(pi z), which vanishes precisely at the integers. Matching these two entire functions of the same zeros and the same growth (and checking one value) forces 1/(Gamma(z)Gamma(1-z)) = sin(pi z)/pi, which is the formula rearranged. The cleanest honest route runs through the Weierstrass product for 1/Gamma and Euler's product for the sine, sin(pi z) = pi z * product over n of (1 - z^2/n^2).

The formula is a workhorse. Set z = 1/2: then Gamma(1/2)^2 = pi/sin(pi/2) = pi, recovering Gamma(1/2) = sqrt(pi) without computing any integral. It also lets you trade a gamma value with argument near a negative number for one with a positive argument, which is invaluable in evaluating integrals and in the functional equation of the zeta function. A subtlety to respect: the formula fails at integers, where sin(pi z) = 0 makes the right side blow up — but that is exactly where one of the two gamma factors has a pole, so both sides are 'infinite' consistently.