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EulerGamma
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EulerGamma

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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)Summary of the most common use cases

Evaluate to any precision:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq-yae
Out[1]=1

Scope  (3)Survey of the scope of standard use cases

Do an exact numerical computation:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq-rix
Out[1]=1

Find decimal digits 10000 through 10050:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq-g3dfe0
Out[1]=1

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq-e5ux47

Applications  (3)Sample problems that can be solved with this function

The first 20 digits of γ in base 10:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1
In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1

Plot scaled sums of divisors:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1

Compute the asymptotic upper bound:

In[2]:=2
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https://wolfram.com/xid/0eny5hueq-d1nyk0
Out[2]=2

Properties & Relations  (2)Properties of the function, and connections to other functions

Various symbolic relations are automatically used:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1

Mathematical functions and operations often give results involving γ:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq-bc3z8
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0eny5hueq-b88cr2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0eny5hueq-eq4pux
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0eny5hueq-i28a3
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0eny5hueq-g80yhs
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0eny5hueq-b4bi8u
Out[6]=6

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1

Neat Examples  (2)Surprising or curious use cases

Terms in the continued fraction:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1

Weyl‐type sum involving EulerGamma:

In[1]:=1
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https://wolfram.com/xid/0eny5hueq
Out[1]=1
Wolfram Research (1988), EulerGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerGamma.html (updated 2007).
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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).

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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.

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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

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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: 06-April-2025 ]}

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@misc{reference.wolfram_2025_eulergamma, author="Wolfram Research", title="{EulerGamma}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/EulerGamma.html}", note=[Accessed: 06-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: 06-April-2025 ]}

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@online{reference.wolfram_2025_eulergamma, organization={Wolfram Research}, title={EulerGamma}, year={2007}, url={https://reference.wolfram.com/language/ref/EulerGamma.html}, note=[Accessed: 06-April-2025 ]}