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Glaisher

Glaisher

is Glaisher's constant with numerical value .

Details

Mathematical constant treated as numeric by NumericQ and as a constant by D.
Glaisher can be evaluated to any numerical precision using N.
Glaisher's constant A satisfies , where is the Riemann zeta function.

Background & Context

Glaisher is the symbol representing Glaisher's constant , also known as the Glaisher–Kinkelin constant. Glaisher has a number of equivalent definitions throughout mathematics but is most commonly defined as the constant that satisfies , where is the Riemann zeta function Zeta, is its derivative evaluated at , and Log is the natural logarithm. Glaisher has a numerical value . Glaisher arises in mathematical computations including sums, products, and integrals but is especially prominent in sums and integrals involving Gamma and Zeta functions.
When Glaisher is used as a symbol, it is propagated as an exact quantity. Expansion and simplification of complicated expressions involving Glaisher may require use of functions such as FunctionExpand and FullSimplify.
It is not currently known if Glaisher is rational (meaning it can be expressed as a ratio of integers), algebraic (meaning it is the root of some integer polynomial), or normal (meaning the digits in its base- expansion are equally distributed) to any base.
Glaisher can be evaluated to arbitrary numerical precision using N. However, no efficient formulas for computing large numbers of its digits are currently known. RealDigits can be used to return a list of digits of Glaisher and ContinuedFraction to obtain terms of its continued fraction expansion.

Examples

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Basic Examples (1)

Evaluate to any precision:

Wolfram Language code: N[Glaisher, 50]

Scope (2)

Do an exact numerical computation:

Wolfram Language code: Round[Glaisher ^ 100]

TraditionalForm formatting:

Wolfram Language code: Glaisher//TraditionalForm

Applications (5)

Derivatives of zeta functions:

Wolfram Language code: Zeta'[-1]

Wolfram Language code: Derivative[1, 0][Zeta][-1, 1 / 4]

Integrals containing gamma functions:

Wolfram Language code: Integrate[LogGamma[t], {t, 0, 1 / 2}]

Get Glaisher from an infinite product:

Wolfram Language code: Product[k ^ ((1/k^2)), {k, 1, Infinity}]

Get Glaisher from infinite sums:

Wolfram Language code: Sum[Log[k] / k ^ 2, {k, Infinity}]

Obtains Glaisher from a limit:

Wolfram Language code: Limit[ (Hyperfactorial[n]/n^n(n + 1) / 2Exp[-n^2 / 4]n^1 / 12), n -> Infinity]

Properties & Relations (1)

Various symbolic relations are automatically used:

Wolfram Language code: 1 < Glaisher < 2

Neat Examples (1)

Terms in the continued fraction:

Wolfram Language code: ListLinePlot[ContinuedFraction[Glaisher, 200], PlotRange -> All]

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