Stieltjes constants
Encyclopedia
In mathematics
, the Stieltjes constants are the numbers
that occur in the Laurent series
expansion of the Riemann zeta function:
The Stieltjes constants are given by the limit
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...
, the Stieltjes constants are the numbers
that occur in the Laurent seriesLaurent series
In mathematics, the Laurent series of a complex function f 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...
expansion of the Riemann zeta function:
The Stieltjes constants are given by the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
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(In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.)
Cauchy's differentiation formulaCauchy's integral formulaIn 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 boundary of the disk, and it provides integral formulas for all...
leads the integral representation
The zero'th constant
is known as the Euler–Mascheroni constant.
The first few values are:
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- {| class="wikitable"
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| n || approximate value of γn || OEIS
|-
| 0 || 0.5772156649015328606065120900824024310421 ||
|-
| 1 || -0.072815845483676724860586 ||
|-
| 2 || -0.0096903631928723184845303 ||
|-
| 3 || 0.002053834420303345866160 ||
|-
| 4 || 0.0023253700654673000574 ||
|-
| 5 || 0.0007933238173010627017 ||
|-
| 6 || -0.00023876934543019960986 ||
|-
| 7 || -0.0005272895670577510 ||
|-
| 8 || -0.00035212335380 ||
|-
| 9 || -0.0000343947744 ||
|-
| 10 || 0.000205332814909 ||
|}
More generally, one can define Stieltjes constants
that occur in the Laurent seriesLaurent seriesIn mathematics, the Laurent series of a complex function f 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...
expansion of the Hurwitz zeta function:
Here q is a complex numberComplex numberA 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(q)>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have
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