Difference between revisions of "Exponential integral Ei"

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[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions]
 
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions]
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Revision as of 22:12, 16 August 2015

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

Properties

Theorem

The following formula holds: $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ where $\mathrm{li}$ denotes the logarithmic integral, $\mathrm{Ei}$ denotes the exponential integral Ei, and $\log$ denotes the logarithm.

Proof

References

Theorem

The following formula holds for $x>0$: $$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!},$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\log$ denotes the logarithm, and $\gamma$ denotes the Euler-Mascheroni constant.

Proof

References

References

Exponential Integral and Related Functions

$\ast$-integral functions