“Dirichlet beta function”的意思、由来-开放百科全书

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Dirichlet beta function

释义

Definition
Euler product formula
Functional equation
Special values
See also
References

{{About|the Dirichlet beta function|other beta functions|Beta function (disambiguation)}}

In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

Definition

The Dirichlet beta function is defined as

or, equivalently,

In each case, it is assumed that Re(s) > 0.

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:

proof

Another equivalent definition, in terms of the Lerch transcendent, is:

which is once again valid for all complex values of s.

Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function

Euler product formula

It is also the simplest example of a series non-directly related to which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.

At least for Re(s) ≥ 1:

where {{math|1=p≡1 mod 4}} are the primes of the form {{math|1=4n+1}} (5,13,17,...) and {{math|1=p≡3 mod 4}} are the primes of the form {{math|1=4n+3}} (3,7,11,...). This can be written compactly as

Functional equation

The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It is given by

where Γ(s) is the gamma function.

Special values

Some special values include:

where G represents Catalan's constant, and

where in the above is an example of the polygamma function. More generally, for any positive integer k:

where represent the Euler numbers. For integer k ≥ 0, this extends to:

Hence, the function vanishes for all odd negative integral values of the argument.

For every positive integer k:

{{cn|date=September 2016}}

where is the Euler zigzag number.

Also it was derived by Malmsten in 1842 that

s	approximate value β(s)	OEIS
1/5	0.5737108471859466493572665	A261624}}
1/4	0.5907230564424947318659591	A261623}}
1/3	0.6178550888488520660725389	A261622}}
1/2	0.6676914571896091766586909	A195103}}
1	0.7853981633974483096156608	A003881}}
2	0.9159655941772190150546035	A006752}}
3	0.9689461462593693804836348	A153071}}
4	0.9889445517411053361084226	A175572}}
5	0.9961578280770880640063194	A175571}}
6	0.9986852222184381354416008	A175570}}
7	0.9995545078905399094963465
8	0.9998499902468296563380671
9	0.9999496841872200898213589
10	0.9999831640261968774055407

There are zeros at -1; -3; -5; -7 etc.

References

{{cite journal

|first1=M. L.
|last1=Glasser
|title=The evaluation of lattice sums. I. Analytic procedures
|year=1972
|journal=J. Math. Phys.
|doi=10.1063/1.1666331
|volume=14
|issue=3
|page=409
|bibcode=1973JMP....14..409G
}}

J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
{{MathWorld|title=Dirichlet Beta Function|urlname=DirichletBetaFunction}}

1 : Zeta and L-functions

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Definition

Euler product formula

Functional equation

Special values

See also

References