Difference between revisions of "Exponential integral Ei"

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(Created page with "The exponential integral $\mathrm{Ei}$ is defined by $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi.$$ =Properties= {{:Relationship between log...")
 
 
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The exponential integral $\mathrm{Ei}$ is defined by
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The exponential integral $\mathrm{Ei}$ is defined for $x>0$ by
$$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi.$$
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$$\mathrm{Ei}(x) = \mathrm{PV}\int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t,$$
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where $\mathrm{PV}$ denotes the [[Cauchy principal value]] and $e^t$ denotes the [[exponential]].
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<div align="center">
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<gallery>
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File:Eiplot.png|Graph of $\mathrm{Ei}$.
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File:Complexeiplot.png|Domain coloring of $\mathrm{Ei}$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
{{:Relationship between logarithmic integral and exponential integral}}
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[[Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt]]<br />
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[[Relationship between logarithmic integral and exponential integral]]<br />
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[[Exponential integral Ei series]]<br />
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[[Relationship between exponential integral Ei, cosine integral, and sine integral]]<br />
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=See Also=
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[[Exponential integral E]]
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=References=
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* {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|next=Ei(-x)=-Integral from x to infinity of e^(-t)/t dt}}
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt|next=Logarithmic integral}}: $5.1.2$
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{{:*-integral functions footer}}
  
{{:Exponential integral Ei series}}
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[[Category:SpecialFunction]]

Latest revision as of 00:48, 24 March 2018

The exponential integral $\mathrm{Ei}$ is defined for $x>0$ by $$\mathrm{Ei}(x) = \mathrm{PV}\int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t,$$ where $\mathrm{PV}$ denotes the Cauchy principal value and $e^t$ denotes the exponential.


Properties

Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt
Relationship between logarithmic integral and exponential integral
Exponential integral Ei series
Relationship between exponential integral Ei, cosine integral, and sine integral

See Also

Exponential integral E

References

$\ast$-integral functions