Dirichlet beta function
The Dirichlet beta function, written by the Greek letter, is a special mathematical function that plays a role in analytic number theory, a branch of mathematics. It is related to the Riemann zeta function.
It was named after the German mathematician Peter Gustav Lejeune Dirichlet ( 1805-1859 ).
Definition
Of a complex number whose real part is larger than 0, the Beta function is defined on the Dirichlet:
Although this expression converges only on the right half-plane, it provides the basis for all other representations of the beta function dar. To calculate the beta function for all numbers in the complex plane one uses its analytic continuation.
Product representation
For the Beta function, there is a product description that converges for all complex, whose real part is greater than 1.
This implies that all primes of the form (ie ) is multiplied. Analogous means that all prime numbers, which have the form (ie, ) is multiplied.
Functional equation
For all the functional equation holds:
Here is the gamma function.
It extends the domain of the beta function on the entire complex plane.
Further illustrations
About the Mellin transform of the function we obtain the integral representation:
Where again denotes the gamma function.
Together with the Hurwitz zeta function are obtained for all the complex relation:
Another equivalent representation for all complex includes the transcendent lerchsche zeta function and reads:
Special values
Some special values of the function are
Herein, the constant and the third catalansche Polygamma function.
In general, for positive integers recursion:
Where the- th Euler number. In case this simplifies to
Moreover, for natural:
Derivation
A derivative term for all is given by:
Special values of the derivative function are:
(see sequence A113847 in OEIS A078127 in OEIS sequence and the Euler - Mascheroni constant).
Also applies to positive integers:
More
In addition Guillera and Sondow proved in 2005 the following formula: