Difference between revisions of "Euler-Mascheroni constant"
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(Created page with "$$\gamma = \lim_{n \rightarrow \infty} 1 + \dfrac{1}{2} + \ldots + \dfrac{1}{n}-\log(n) = 0.577215664901532$$") |
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| − | $$\gamma = \lim_{n \rightarrow \infty} | + | __NOTOC__ |
| + | The Euler-Mascheroni constant is the number $\gamma$ defined by the formula | ||
| + | $$\gamma = \lim_{n \rightarrow \infty} H_n-\log(n) = 0.577215664901532 \ldots,$$ | ||
| + | where $H_n$ denotes the $n$th [[harmonic number]]. | ||
| + | |||
| + | =Properties= | ||
| + | [[The Euler-Mascheroni constant exists]]<br /> | ||
| + | [[Reciprocal gamma written as an infinite product]]<br /> | ||
| + | [[Exponential integral Ei series]]<br /> | ||
| + | |||
| + | =Further properties= | ||
| + | The Euler-Mascheroni constant appears in the definition of... | ||
| + | #the [[hyperbolic cosine integral]] | ||
| + | #the [[Barnes G]] function | ||
| + | |||
| + | =See Also= | ||
| + | [[Meissel-Mertens constant]] | ||
| + | |||
| + | =External links= | ||
| + | [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Collection of formulae for Euler's constant g]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|author2=George Neville Watson|edpage=Third edition|prev=Gamma|next=Reciprocal gamma written as an infinite product}}: $\S 12 \cdot 1$ | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Reciprocal gamma written as an infinite product|next=findme}}: §1.1 (4) | ||
| + | * {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Harmonic number}}: $7.(1)$ | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Limit of x^a log(x)=0|next=x/(1+x) less than log(1+x)}}: $4.1.32$ | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Reciprocal gamma written as an infinite product|next=findme}}: 6.1.3 | ||
Latest revision as of 20:57, 3 March 2018
The Euler-Mascheroni constant is the number $\gamma$ defined by the formula $$\gamma = \lim_{n \rightarrow \infty} H_n-\log(n) = 0.577215664901532 \ldots,$$ where $H_n$ denotes the $n$th harmonic number.
Properties
The Euler-Mascheroni constant exists
Reciprocal gamma written as an infinite product
Exponential integral Ei series
Further properties
The Euler-Mascheroni constant appears in the definition of...
- the hyperbolic cosine integral
- the Barnes G function
See Also
External links
Collection of formulae for Euler's constant g
References
- 1920: Edmund Taylor Whittaker and George Neville Watson: A course of modern analysis ... (previous) ... (next): $\S 12 \cdot 1$
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (4)
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $7.(1)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.32$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.1.3
