Exponential integral Ei

The exponential integral $\mathrm{Ei}$ is defined by $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t, \quad \left|\mathrm{arg}(-z) \right|<\pi.$$

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  Graph of $\mathrm{Ei}$.

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  Domain coloring of $\mathrm{Ei}$.

Properties

Relationship between logarithmic integral and exponential integral
Exponential integral Ei series
Relationship between exponential integral Ei, cosine integral, and sine integral

References

Exponential Integral and Related Functions
On certain definite integrals involving the exponential-integral - J.W.L. Glaisher

$\ast$-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