Difference between revisions of "Exponential integral E"
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| − | The exponential integral functions $E_n$ are defined for $\left|\mathrm{arg \hspace{2pt}}z\right|<\pi$ | + | 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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$$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$ | $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$ | ||
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=Properties= | =Properties= | ||
| − | [[Relationship between the exponential integral and upper incomplete gamma function]] | + | [[Relationship between the exponential integral and upper incomplete gamma function]]<br /> |
| + | [[Symmetry relation of exponential integral E]]<br /> | ||
| + | [[Recurrence relation of exponential integral E]]<br /> | ||
=Videos= | =Videos= | ||
| − | [https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br /> | + | [https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral (2 January 2015)]<br /> |
=See Also= | =See Also= | ||
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=References= | =References= | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next= | + | * {{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>) |
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=findme}}: $5.1.4$ | + | * {{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 $n=0,1,2,\ldots$) |
{{:*-integral functions footer}} | {{:*-integral functions footer}} | ||
[[Category:SpecialFunction]] | [[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 $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$
Graph of $\mathrm{E}_1$.
Graph of $\mathrm{E}_2$.
Graph of $\mathrm{E}_3$.
Domain coloring of $\mathrm{E}_1$.
Domain coloring of $\mathrm{E}_2$.
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
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.1$ (note: this formula only defines it for $n=1$)
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.4$ (note: this formula defines it for $n=0,1,2,\ldots$)
