Hadjicostas's formula
Encyclopedia
In mathematics
, Hadjicostas's formula is a formula relating a certain double integral
to values of the Gamma function
and the Riemann zeta function.
with Re(s) > −2. Then
Here
is the Gamma function
and
is the Riemann zeta function.
to get the full result.
as a double integral by letting s tend to −1:
The latter formula was first discovered by Jonathan Sondow and is the one referred to in the title of Hadjicostas's paper.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Hadjicostas's formula is a formula relating a certain double integral
Multiple integral
The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...
to values of the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and the Riemann zeta function.
Statement
Let s be a complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
with Re(s) > −2. Then
Here
is the Gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and
is the Riemann zeta function.Background
The first instance of the formula was proved and used by Frits Beukers in his 1978 paper giving an alternative proof of Apéry's theorem. He proved the formula when s = 0, and proved an equivalent formulation for the case s = 1. This led Petros Hadjicostas to conjecture the above formula in 2004, and within a week it had been proven by Robin Chapman. He proved the formula holds when Re(s) > −1, and then extended the result by analytic continuationAnalytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
to get the full result.
Special cases
As well as the two cases used by Beukers to get alternate expressions for ζ(2) and ζ(3), the formula can be used to express the Euler-Mascheroni constantEuler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
as a double integral by letting s tend to −1:
The latter formula was first discovered by Jonathan Sondow and is the one referred to in the title of Hadjicostas's paper.
See also
- Sondow, J. (2005). "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula," American Mathematical MonthlyAmerican Mathematical MonthlyThe American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
112: 61-65.