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Euler's constant (Definition)

Euler's constant $ \gamma$ is defined by

$\displaystyle \gamma = \lim_{n\rightarrow \infty}\; \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln {n}\right) $

or equivalently

$\displaystyle \gamma = \lim_{n\rightarrow \infty}\; \sum_{i=1}^n \left[\frac{1}{i} - \ln \left( 1 + \frac{1}{i} \right) \right] $

Euler's constant has the value

$\displaystyle 0.57721566490153286060651209008240243104\ldots $

It is related to the gamma function by

$\displaystyle \gamma = - \Gamma'(1) $

It is not known whether $ \gamma$ is rational or irrational.

References.



"Euler's constant" is owned by akrowne.
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Other names:  Euler-Mascheroni constant, Mascheroni constant

Attachments:
integral representations of the Mascheroni constant (Theorem) by rspuzio
proof that Euler's constant exists (Proof) by rm50
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Cross-references: irrational, rational, gamma function
There are 19 references to this entry.

This is version 7 of Euler's constant, born on 2002-02-10, modified 2006-06-26.
Object id is 1883, canonical name is EulersConstant.
Accessed 16792 times total.

Classification:
AMS MSC40A25 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Approximation to limiting values )

Pending Errata and Addenda
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