EulerGamma

EulerGamma

is Euler’s constant γ, with numerical value .

Details

  • Mathematical constant treated as numeric by NumericQ and as a constant by D.
  • EulerGamma can be evaluated to any numerical precision using N.

Background & Context

  • EulerGamma is the symbol representing Euler's constant γ, which is also known as the Euler–Mascheroni constant. EulerGamma has a number of equivalent definitions in mathematics but is most commonly defined as the limiting value lim_(n->infty) (TemplateBox[{n}, HarmonicNumber]-log(n)) involving HarmonicNumber[n] and the natural logarithm Log[n]. EulerGamma has a numerical value . EulerGamma arises in mathematical computations including sums, products, integrals, and limits.
  • When EulerGamma is used as a symbol, it is propagated as an exact quantity. Expansion and simplification of complicated expressions involving EulerGamma may require use of functions such as FunctionExpand and FullSimplify.
  • Most mathematicians believe EulerGamma is not only irrational (meaning it cannot be expressed as a ratio of any two integers) but also transcendental (meaning it is not the root of any integer polynomial). lt is also not known if EulerGamma is normal (meaning the digits in its base-b expansion are equally distributed) to any base. Despite its appearance in various closed-form sums and integrals, EulerGamma is conjectured to not be a Kontsevich–Zagier period (meaning it is not the value of an absolutely convergent integral of any univariate or multivariate rational function with rational coefficients over algebraically-specified domains in ). However, all of the above conjectures currently remain unproved.
  • EulerGamma can be evaluated to arbitrary numerical precision using N. In fact, calculating the first decimal digits of EulerGamma takes only a fraction of a second on a modern desktop computer. RealDigits can be used to return a list of digits of EulerGamma and ContinuedFraction to obtain terms of its continued fraction expansion.
  • The Stieltjes constants StieltjesGamma[n] with TemplateBox[{0}, StieltjesGamma]=ℽ generalize EulerGamma.

Examples

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Basic Examples  (1)

Evaluate to any precision:

Scope  (3)

Do an exact numerical computation:

Find decimal digits 10000 through 10050:

TraditionalForm formatting:

Applications  (3)

The first 20 digits of γ in base 10:

Plot scaled sums of divisors:

Compute the asymptotic upper bound:

Properties & Relations  (2)

Various symbolic relations are automatically used:

Mathematical functions and operations often give results involving γ:

Possible Issues  (1)

It is currently not known if EulerGamma is an algebraic number:

Neat Examples  (2)

Terms in the continued fraction:

Weyl‐type sum involving EulerGamma:

Wolfram Research (1988), EulerGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerGamma.html (updated 2007).

Text

Wolfram Research (1988), EulerGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerGamma.html (updated 2007).

CMS

Wolfram Language. 1988. "EulerGamma." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/EulerGamma.html.

APA

Wolfram Language. (1988). EulerGamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerGamma.html

BibTeX

@misc{reference.wolfram_2025_eulergamma, author="Wolfram Research", title="{EulerGamma}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/EulerGamma.html}", note=[Accessed: 27-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_eulergamma, organization={Wolfram Research}, title={EulerGamma}, year={2007}, url={https://reference.wolfram.com/language/ref/EulerGamma.html}, note=[Accessed: 27-April-2025 ]}