Difference between revisions of "Exponential integral E"

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The exponential integrals are
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__NOTOC__
$$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi,$$
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The exponential integral functions $E_n$ are defined for $\left|\mathrm{arg \hspace{2pt}}z\right|<\pi$ and $n=1,2,3,\ldots$ by
$$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z<\pi|,$$
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$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$
and
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$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} dt.$$
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<div align="center">
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<gallery>
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File:E1plot.png|Graph of $\mathrm{E}_1$.
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File:E2plot.png|Graph of $\mathrm{E}_2$.
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File:E3plot.png|Graph of $\mathrm{E}_3$.
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File:Complexe1plot.png|[[Domain coloring]] of $\mathrm{E}_1$.
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File:Complexe2plot.png|[[Domain coloring]] of $\mathrm{E}_2$.
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</gallery>
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</div>
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=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Relationship between the exponential integral and upper incomplete gamma function]]<br />
<strong>Proposition:</strong> The exponential integral $\mathrm{Ei}$ is related to the [[logarithmic integral]] by the formula
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[[Symmetry relation of exponential integral E]]<br />
$$\mathrm{li}(x)=\mathrm{Ei}( \log(x)); n=0,1,2,\ldots, \mathrm{Re}(z)>0$$
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[[Recurrence relation of exponential integral E]]<br />
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>
 
</div>
 
</div>
 
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br />
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[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral (2 January 2015)]<br />
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=See Also=
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[[Exponential integral Ei]]
  
 
=References=
 
=References=
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions]
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt}}: $5.1.1$ (<i>note: this formula only defines it for $n=1$</i>)
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=findme}}: $5.1.4$ (<i>note:</i> this formula defines it for $n=0,1,2,\ldots$)
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{{:*-integral functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 00:45, 24 March 2018

The exponential integral functions $E_n$ are defined for $\left|\mathrm{arg \hspace{2pt}}z\right|<\pi$ and $n=1,2,3,\ldots$ by $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$

Properties

Relationship between the exponential integral and upper incomplete gamma function
Symmetry relation of exponential integral E
Recurrence relation of exponential integral E

Videos

Laplace transform of exponential integral (2 January 2015)

See Also

Exponential integral Ei

References

$\ast$-integral functions