Confluent Hypergeometric Function - Application To Continued Fractions

Application To Continued Fractions

By applying a limiting argument to Gauss's continued fraction it can be shown that

\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}} {1 + \cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}} {1 - \cfrac{{\displaystyle\frac{b-a+1}{(b+2)(b+3)}z}} {1 + \cfrac{{\displaystyle\frac{a+2}{(b+3)(b+4)}z}}{1 - \ddots}}}}}

and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole.

Read more about this topic:  Confluent Hypergeometric Function

Famous quotes containing the words application to, application and/or continued:

    “Preaching is the expression of the moral sentiment in application to the duties of life.”
    —Ralph Waldo Emerson (1803–1882)

    “By an application of the theory of relativity to the taste of readers, to-day in Germany I am called a German man of science, and in England I am represented as a Swiss Jew. If I come to be regarded as a bête noire the descriptions will be reversed, and I shall become a Swiss Jew for the Germans and a German man of science for the English!”
    —Albert Einstein (1879–1955)

    “If my sons are to become the kind of men our daughters would be pleased to live among, attention to domestic details is critical. The hostilities that arise over housework...are crushing the daughters of my generation....Change takes time, but men’s continued obliviousness to home responsibilities is causing women everywhere to expire of trivialities.”
    —Mary Kay Blakely (20th century)