Exponential integral Ei

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.

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  Graph of $\mathrm{Ei}$.

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  Domain coloring of $\mathrm{Ei}$.

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

• James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (next)
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.2$

$\ast$-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