|
In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if ''n'' is a positive integer: :. The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral: :. This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function: :. The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics. == Motivation == The gamma function can be seen as a solution to the following interpolation problem: : "Find a smooth curve that connects the points (''x'', ''y'') given by ''y'' = (''x'' − 1)! at the positive integer values for ''x''." 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, ''x''! = 1 × 2 × ... × ''x'', 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; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express ''x''!. Stirling's approximation is asymptotically equal to the factorial function for large values of ''x''. It is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function. 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 (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, such as ''k'' sin ''n'' , will give another function with that property. A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, : for ''x'' equal to any positive real number. The Bohr–Mollerup theorem proves that these properties, together with the assumption that ''f'' be logarithmically convex (or "superconvex"〔Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.〕), uniquely determine ''f'' for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of ''f''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「gamma function」の詳細全文を読む スポンサード リンク
|