Difference between revisions of "Exponential integral E"

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The exponential integral functions $E_n$ are defined by
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$$E_1(z) = \displaystyle\int_1^{\infty} \dfrac{e^{-t}}{t} \mathrm{d}t, \quad \left|\mathrm{arg \hspace{2pt}}z\right|<\pi,$$
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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
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$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$
 
$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$
  
=Properties=
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<gallery>
<strong>Theorem:</strong> The exponential integral $E_1$ has series representation
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File:E1plot.png|Graph of $\mathrm{E}_1$.
$$E_1(z)=-\gamma-\log z - \displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^kz^k}{kk!}, \quad |\mathrm{arg}(z)|<\pi,$$
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File:E2plot.png|Graph of $\mathrm{E}_2$.
where $\gamma$ denotes the [[Euler-Mascheroni constant]].
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File:E3plot.png|Graph of $\mathrm{E}_3$.
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File:Complexe1plot.png|[[Domain coloring]] of $\mathrm{E}_1$.
<strong>Proof:</strong> █
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File:Complexe2plot.png|[[Domain coloring]] of $\mathrm{E}_2$.
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</gallery>
 
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=Properties=
<strong>Theorem (Symmetry):</strong> The following symmetry relation holds:
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[[Relationship between the exponential integral and upper incomplete gamma function]]<br />
$$E_n(\overline{z})=\overline{E_n(z)}.$$
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[[Symmetry relation of exponential integral E]]<br />
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[[Recurrence relation of exponential integral E]]<br />
<strong>Proof:</strong>
 
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=Videos=
<strong>Theorem (Recurrence):</strong> The following recurrence holds:
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[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral (2 January 2015)]<br />
$$E_{n+1}(z) = \dfrac{1}{n}[e^{-z}-zE_n(z)];(n=1,2,3,\ldots).$$
 
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<strong>Proof:</strong> █
 
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=See Also=
<strong>Theorem ([[Continued fraction]]):</strong> The following formula holds:
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[[Exponential integral Ei]]
$$E_n(z)=e^{-z} \left( \dfrac{1}{z+} \dfrac{n}{1+} \dfrac{1}{z+} \dfrac{n+1}{1+} \dfrac{2}{z+} \ldots \right); |\mathrm{arg} z|<\pi.$$
 
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<strong>Proof:</strong> █
 
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=References=
<strong>Theorem:</strong> The following value is known:
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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>)
$$E_n(0)=\dfrac{1}{n-1}; n>1.$$
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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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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following closed form expression is known:
 
$$E_0(z)=\dfrac{e^{-z}}{z}.$$
 
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<strong>Proof:</strong> █
 
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{{:*-integral functions footer}}
<strong>Theorem (Derivative):</strong> $$\dfrac{d}{dz} E_n(z) = -E_{n-1}(z); n=1,2,3,\ldots$$
 
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<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
{{:Relationship between the exponential integral and upper incomplete gamma function}}
 
 
 
=Videos=
 
[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br />
 
 
 
=References=
 
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions]
 
  
<center>{{:*-integral functions footer}}</center>
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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