Computing Gamma Values

Complex Analysis · Axiom Academy

EXAMPLE Computing Gamma Values Evaluating at a half-integer with the recurrence and the anchor value (1/2) = Compute the exact value of . You may use the recurrence and the special value . Each step of the recurrence peels off one factor and lowers the argument by 1 . Stepping down from lands on the known anchor . Nice work — you computed by walking the recurrence down to a single known value. The recurrence is the engine: lets you trade any half-integer argument for one that is 1 smaller, peeling off a factor each time. Anchor on a known value: for half-integers everything reduces to (which comes from the Gaussian integral ). Run it backward for negatives: rewriting the recurrence as gives values like . Two facts — a recurrence and one anchor — pin down at every half-integer, the same machinery that extends the factorial to the whole complex plane.

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