Beta function

The Euler beta function, also Eulerian integral of first kind (after Leonhard Euler ) is a mathematical function of two complex numbers is denoted by. Its definition is:

Must have being and a positive real part.

The beta function was the first known scattering amplitude in string theory. It occurs in addition to in the beta distribution.

General

For a fixed (or ) is a holomorphic function of (respectively), and for the function of the symmetry relation holds

There are these integral representations for the beta function and

The main result of the theory of beta function is the identity

Where is the Euler gamma function called. At this presentation can also be seen that the analytic continuation of the beta function has poles precisely along and integers.

Theodor Schneider showed in 1940 that for all rational, non-integer x, y is transcendental.

Representations

The beta function has many other representations, such as:

Beta function may be used by adjusting the indices defining the binomial coefficients:

With the representation for the gamma function to get for integer and:

Derivation

The derivative is given by

The digamma function is.