Difference between revisions of "Euler-Mascheroni constant"

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$$\gamma = \lim_{m \rightarrow \infty} 1 + \dfrac{1}{2} + \ldots + \dfrac{1}{m}-\log(m) = 0.577215664901532 \ldots$$
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__NOTOC__
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The Euler-Mascheroni constant is the number $\gamma$ defined by the formula
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$$\gamma = \lim_{n \rightarrow \infty} H_n-\log(n) = 0.577215664901532 \ldots,$$
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where $H_n$ denotes the $n$th [[harmonic number]].
  
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<div class="toccolours mw-collapsible mw-collapsed">
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=Properties=
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<strong>Theorem:</strong> The limit defining $\gamma$ exists.
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[[The Euler-Mascheroni constant exists]]<br />
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<div class="mw-collapsible-content">
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[[Reciprocal gamma written as an infinite product]]<br />
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<strong>Proof:</strong> Let $u_n=\displaystyle\int_0^1 \dfrac{t}{n(t+n)} dt$. Clearly,
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[[Exponential integral Ei series]]<br />
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$$0 < u_n < \displaystyle\int_0^1 \dfrac{1}{n^2} dt = \dfrac{1}{n^2}.$$
 
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Now compute
 
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$$\begin{array}{ll}
 
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u_n &= \dfrac{1}{n} \displaystyle\int_0^1 1 - \dfrac{n}{t+n} dt \\
 
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&= \dfrac{1}{n} \left[ 1 - n(\log (1+n)-\log n) \right] \\
 
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&= \dfrac{1}{n} - \log \left(\dfrac{n+1}{n} \right).
 
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\end{array}$$
 
  
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Since $u_n < \dfrac{1}{n^2}$ and we know that $\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^2}$ converges (it is the [[Riemann zeta]] function evaluated at $z=2$), we can conclude that $\displaystyle\sum_{k=1}^{\infty} u_k$ also converges.
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=Further properties=
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The Euler-Mascheroni constant appears in the definition of...
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#the [[hyperbolic cosine integral]]
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#the [[Barnes G]] function
  
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Notice that due to [[telescoping]] and the properties of the [[logarithm]],
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=See Also=
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$$\displaystyle\sum_{k=1}^m \log \left( \dfrac{k+1}{k} \right) = \log \left(\dfrac{2}{1} \right) + \log\left( \dfrac{3}{2} \right) + \ldots \log \left( \dfrac{m+1}{m} \right)= \log(m+1)$$
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[[Meissel-Mertens constant]]
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Now we see
 
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$$\begin{array}{ll}
 
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\displaystyle\sum_{k=1}^{\infty} u_k &= \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k} - \log \left( \dfrac{k+1}{k} \right) \\
 
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&=\displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^{m} \dfrac{1}{k} - \log(m+1) \right].
 
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\end{array}$$
 
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Since $\displaystyle\lim_{m \rightarrow \infty} \log \left( \dfrac{m+1}{m} \right)=0$, we may rewrite it as $\log(m+1)-\log(m)$ and insert it into the above equation to get
 
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$$\begin{array}{ll}
 
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\displaystyle\sum_{k=1}^{\infty} u_k &=\displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^{m} \dfrac{1}{k} - \log(m+1) + \log(m+1)-\log(m) \right] \\
 
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&= \displaystyle\lim_{m \rightarrow \infty} \left( 1 + \dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{m} - \log(m) \right),
 
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\end{array}$$
 
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as was to be shown. █
 
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</div>
 
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</div>
 
  
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=Properties=
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=External links=
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{{:Gamma function Weierstrass product}}
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[http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Collection of formulae for Euler's constant g]<br />
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=References=
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* {{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}}: $\S 12 \cdot 1$
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* {{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)
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* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Harmonic number}}: $7.(1)$
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* {{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)}}: $4.1.32$
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* {{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...

  1. the hyperbolic cosine integral
  2. the Barnes G function

See Also

Meissel-Mertens constant

External links

Collection of formulae for Euler's constant g

References

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