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 and/or function:

    “... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
    —Adrienne Rich (b. 1929)

    “The mother’s and father’s attitudes toward the child correspond to the child’s own needs.... Mother has the function of making him secure in life, father has the function of teaching him, guiding him to cope with those problems with which the particular society the child has been born into confronts him.”
    —Erich Fromm (1900–1980)