Difference between revisions of "Exponential integral E"

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(Created page with "The exponential integrals are $$\mathrm{Ei}(x) = \int_{-\infty}^x \dfrac{e^t}{t} dt$$ and $$E_1(x) = \int_x^{\infty} \dfrac{e^{-t}}{t} dt.$$ Simple properties of integrals imp...")
 
 
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The exponential integrals are
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__NOTOC__
$$\mathrm{Ei}(x) = \int_{-\infty}^x \dfrac{e^t}{t} dt$$
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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
and
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$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$
$$E_1(x) = \int_x^{\infty} \dfrac{e^{-t}}{t} dt.$$
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Simple properties of integrals imply that $E_1(x) = -\mathrm{Ei}(-x)$.
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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=
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[[Relationship between the exponential integral and upper incomplete gamma function]]<br />
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[[Symmetry relation of exponential integral E]]<br />
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[[Recurrence relation of exponential integral E]]<br />
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=Videos=
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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]]
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=References=
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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