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(Also known as the ''Euler-Mascheroni'' constant, cite Finch (2003))
(moved text from Catalan constant)
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The number $\gamma$ defined by
 
The number $\gamma$ defined by
  
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====Comments====
 
====Comments====
Also known as the ''Euler-Mascheroni'' constant.
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Also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
  
 
====References====
 
====References====
 
{|
 
{|
 
|-
 
|-
|valign="top"|{{Ref|Fi}}||valign="top"|  Steven R. Finch,  "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press  (2003)  ISBN 0-521-81805-2 Zbl 1054.00001
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|valign="top"|{{Ref|Fi}}||valign="top"|  Steven R. Finch,  "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press  (2003)  ISBN 0-521-81805-2 {{ZBL|1054.00001}}
 
|-
 
|-
 
|}
 
|}
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====Comments====
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==Euler–Mascheroni constant.==
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Indeed, one also has
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004038.png" /></td> </tr></table>
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and
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004040.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004041.png" /></td> </tr></table>
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where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004042.png" />, Euler's classical results state:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004044.png" /></td> </tr></table>
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Erdélyi,  W. Magnus,  F. Oberhettinger,  F.G. Tricomi,  "Higher transcendental functions" , '''I''' , McGraw-Hill  (1953)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Lewin,  "Polylogarithms and associated functions" , Elsevier  (1981)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  H.M. Srivastava,  J. Choi,  "Series associated with the zeta and related functions" , Kluwer Acad. Publ.  (2001)</TD></TR>
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</table>

Revision as of 19:09, 29 December 2014

The number $\gamma$ defined by

$$ \gamma=\lim_{n\to \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\right)\approx 0.57721566490\ldots,$$

considered by L. Euler (1740). Its existence follows from the fact that the sequence

$$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$

is monotone increasing and bounded from above. The number-theoretic nature of the Euler constant has not been studied; it is not even known (2012) whether it is a rational number or not.

In fact, a relation

$$ \sum_{n\leq x}\,\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$

holds, cf. [HaWr, Chapter 22.5].

References

[HaWr] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001

Comments

Also known as the Euler-Mascheroni constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).

References

[Fi] Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001

Comments

Euler–Mascheroni constant.

Indeed, one also has

(a13)

and

(a14)

where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function , Euler's classical results state:

(a15)

References

[a1] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , I , McGraw-Hill (1953)
[a2] L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981)
[a3] H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001)
How to Cite This Entry:
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=35954
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article