Fungsi gamma: Perbedaan antara revisi
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Motivasi - minor |
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Baris 22:
A 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 simple formula for the factorial, ''n''<nowiki>!</nowiki> = 1 × 2 × … × ''n'', cannot be used directly for fractional values of ''x'' since it is only valid when ''x'' is a [[natural number]] (''i.e.'', a positive integer).
There are, relatively speaking, no such simple solutions for factorials; any combination of sums, products, powers, [[exponential function]]s, or [[logarithm]]s with a fixed number of terms will not suffice to express ''x''<nowiki>!</nowiki>. [[Stirling's approximation]] is asymptotically equal to the factorial function for large values of x.
There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being [[analytic function|analytic]] (except at the non-positive integers), and it can be characterized in several ways. 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 will give another function with that property.
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