Difference between revisions of "Exponential integral Ei series"

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==Theorem==
<strong>[[Exponential integral Ei series|Theorem]]:</strong> The following formula holds:
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The following formula holds for $x>0$:
$$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!}; x>0,$$
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$$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!},$$
 
where $\mathrm{Ei}$ denotes the [[exponential integral Ei]], $\log$ denotes the [[logarithm]], and $\gamma$ denotes the [[Euler-Mascheroni constant]].
 
where $\mathrm{Ei}$ denotes the [[exponential integral Ei]], $\log$ denotes the [[logarithm]], and $\gamma$ denotes the [[Euler-Mascheroni constant]].
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<strong>Proof:</strong>
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==Proof==
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==References==
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* {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|prev=Ei(-x)=-Integral from x to infinity of e^(-t)/t dt|next=Logarithmic integral}} (<i>note: expresses the logarithm term as $\frac{1}{4}\log(x^4)$</i>)
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 03:31, 17 March 2018

Theorem

The following formula holds for $x>0$: $$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!},$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\log$ denotes the logarithm, and $\gamma$ denotes the Euler-Mascheroni constant.

Proof

References