mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Hadjicostas's formula is a formula relating a certain

double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...

to values of the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

and the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

. It is named after Petros Hadjicostas.

Statement

Let ''s'' be a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

with ''s'' ≠ -1 and Re(''s'') > −2. Then :

\int_0^1\int_0^1 \frac(-\log(xy))^s\,dx\,dy=\Gamma(s+2)\left(\zeta(s+2)-\frac\right).

Here Γ is the

Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

and ζ is the

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 In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number :\zeta(3) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots = 1.2020569\ldots cannot be written as a fract ...

. 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 continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...

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 constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...

as a double integral by letting ''s'' tend to −1: :

\gamma=\int_0^1\int_0^1\frac\,dx\,dy.

The latter formula was first discovered by Jonathan Sondow and is the one referred to in the title of Hadjicostas's paper.

Statement

Background

Special cases

Notes

See also