Beta Function - Relationship Between Gamma Function and Beta Function

Relationship Between Gamma Function and Beta Function

To derive the integral representation of the beta function, write the product of two factorials as

\Gamma(x)\Gamma(y) = \int_0^\infty\ e^{-u} u^{x-1}\,du \int_0^\infty\ e^{-v} v^{y-1}\,dv =\int_0^\infty\int_0^\infty\ e^{-u-v} u^{x-1}v^{y-1}\,du \,dv. \!

Changing variables by putting u=zt, v=z(1-t) shows that this is

\int_{z=0}^\infty\int_{t=0}^1\ e^{-z} (zt)^{x-1}(z(1-t))^{y-1}z\,dt \,dz =\int_{z=0}^\infty \ e^{-z}z^{x+y-1} \,dz\int_{t=0}^1t^{x-1}(1-t)^{y-1}\,dt. \!

Hence

\Gamma(x)\,\Gamma(y)=\Gamma(x+y)\Beta(x,y) .

The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking

and, one has:
.

Read more about this topic:  Beta Function

Famous quotes containing the words relationship between, relationship and/or function:

    “Living in cities is an art, and we need the vocabulary of art, of style, to describe the peculiar relationship between man and material that exists in the continual creative play of urban living. The city as we imagine it, then, soft city of illusion, myth, aspiration, and nightmare, is as real, maybe more real, than the hard city one can locate on maps in statistics, in monographs on urban sociology and demography and architecture.”
    —Jonathan Raban (b. 1942)

    “It was a real treat when he’d read me Daisy Miller out loud. But we’d reached the point in our relationship when, in a straight choice between him and Henry James, I’d have taken Henry James any day even if Henry James were dead and not much of a one for the girls when living, either.”
    —Angela Carter (1940–1992)

    “The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
    —Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)