Exponential integral E

The exponential integral functions $E_n$ are defined for $\left|\mathrm{arg \hspace{2pt}}z\right|<\pi$ by $$E_1(z) = \displaystyle\int_1^{\infty} \dfrac{e^{-t}}{t} \mathrm{d}t,$$ and $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} \mathrm{d}t.$$

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  Graph of $\mathrm{E}_1$.

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  Graph of $\mathrm{E}_2$.

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  Graph of $\mathrm{E}_3$.

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  Domain coloring of $\mathrm{E}_1$.

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  Domain coloring of $\mathrm{E}_2$.

Properties

Relationship between the exponential integral and upper incomplete gamma function

Videos

Laplace transform of exponential integral

References

Exponential Integral and Related Functions

$\ast$-integral functions

Cosine integral

Cosh integral

Exp integral $E_n$

Exp integral $\mathrm{Ei}$

Fresnel $C$

Fresnel $S$

Gudermannian

Inverse $\mathrm{gd}$

Sine integral

Log integral

Sinh integral