However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as k sin mπx for an integer m, will give another function with that property. The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. A good solution to this is the gamma function. There are, relatively speaking, no such simple solutions for factorials no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x! but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus. The simple formula for the factorial, x! = 1 × 2 × ⋯ × x, cannot be used directly for fractional values of x since it is only valid when x is a natural number (or positive integer).
"Find a smooth curve that connects the points ( x, y) given by y = ( x − 1)! at the positive integer values for x."Ī plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x. The gamma function can be seen as a solution to the following interpolation problem: The gamma function interpolates the factorial function to non-integer values.
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