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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.31 The Gamma Function

Ben Green

If n is a positive integer, then its factorial, written n!, is the number 1 × 2 × · · · × n: that is, the product of all positive integers up to n. For example, the first eight factorials are 1, 2, 6, 24, 120, 720, 5040, and 40320. (The exclamation mark was introduced by Christian Kramp 200 years ago as a convenience to the printer: it is perhaps also intended to convey some alarm at the rapidity with which n! grows. An obsolete notation, which can still be found in some twentieth-century texts, is Image.) From this definition, it might appear to be impossible to make sense of the idea of the factorial of a number ...

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