![gamma-area](https://upload.wikimedia.org/wikipedia/commons/0/04/Gamma-area.svg)
In
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 ...
, the Stieltjes constants are the numbers
that occur in the
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
expansion of 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) > ...
:
:
The constant
is known as the
Euler–Mascheroni constant.
Representations
The Stieltjes constants are given by the
limit
Limit or Limits may refer to:
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* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
(In the case ''n'' = 0, the first summand requires evaluation of
00, which is taken to be 1.)
Cauchy's differentiation formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundar ...
leads to the integral representation
:
Various representations in terms of integrals and infinite series are given in works of
Jensen, Franel,
Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
,
Hardy,
Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors.
In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
:
where δ
''n,k'' is the
Kronecker symbol (Kronecker delta).
Among other formulae, we find
:
:
see.
As concerns series representations, a famous series implying an integer part of a logarithm was given by
Hardy in 1912
:
Israilov
gave semi-convergent series in terms of
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
:
Connon, Blagouchine
and Coppo
gave several series with the
binomial coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
:
where ''G''
''n'' are
Gregory's coefficients, also known a
reciprocal logarithmic numbers(''G''
1=+1/2, ''G''
2=−1/12, ''G''
3=+1/24, ''G''
4=−19/720,... ).
More general series of the same nature include these examples
:
and
:
or
:
where are the
Bernoulli polynomials of the second kind
The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function:
:
\frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, -1
and
:\gamma=\sum_^\infty\frac\B ...
and are the polynomials given by the generating equation
:
respectively (note that ).
Oloa and Tauraso showed that series with
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
s may lead to Stieltjes constants
:
Blagouchine
obtained slowly-convergent series involving unsigned
Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...