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In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. (The term "confluent" refers to the merging of singular points of families of differential equations; "confluere" is Latin for "to flow together".) There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. * Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation. * Coulomb wave functions are solutions to the Coulomb wave equation. The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables. ==Kummer's equation== Kummer's equation may be written as: : with a regular singular point at and an irregular singular point at . It has two (usually) linearly independent solutions and . Kummer's function (of the first kind) ''M'' is a generalized hypergeometric series introduced in , given by: : is the rising factorial. Another common notation for this solution is . Considered as a function of ''a'', ''b'', or with the other two held constant, this defines an entire function of ''a'' or ''z'', except when As a function of it is analytic except for poles at the non-positive integers. Some values of ''a'' and yield solutions that can be expressed in terms of other known functions. See #Special cases. When ''a'' is a non-positive integer then Kummer's function (if it is defined) is a (generalized) Laguerre polynomial. Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function : This is undefined for integer , but can be extended to integer by continuity. Unlike Kummer's function which is an entire function of ''z'', ''U''(''z'') usually has a singularity at zero. But see #Special cases for some examples where it is an entire function (polynomial). Note that if : which can occur if is a non-positive integer, then and are not independent and another solution is needed. Also when is a non-positive integer we need another solution because then is not defined. For instance, if , Kummer's function is undefined, but two independent solutions are and For ''a'' = 0 but at other values of ''b'', we have the two solutions: : : When ''b'' = 1 this second solution is the exponential integral Ei(''z''). See #Special cases for solutions to some other cases. 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「confluent hypergeometric function」の詳細全文を読む スポンサード リンク
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