Exponential Integral

\mbox{Ei}(x)=-\int_{-x}^{\infty} rac{e^{-t}}{t}\, dt\,.

Since 1/''t'' diverges at ''t'' = 0, the above integral has to be understood in terms of the Cauchy Principal Value .

The exponential integral has the series representation:

:

\mbox{Ei}(x) = \gamma+\ln x+ 
  \sum_{k=1}^{\infty} rac{x^k}{k\; k!} \,,

where γ is the Euler Gamma Constant .

The exponential integral is closely related to the Logarithmic Integral Function li(''x''),

:li(''x'') = Ei (ln (''x'')) for all positive real ''x'' ≠ 1.

Also closely related is a function which integrates over a different range:

:

{
m E}_1(x) = \int_x^\infty rac{e^{-t}}{t}\, dt.

This function may be regarded as extending the exponential integral to the
negative reals by

:

{
m Ei}(-x) = - {
m E}_1(x).\,

We can express both of them in terms of an Entire Function ,

:

{
m Ein}(x) = \int_0^x (1-e^{-t})rac{dt}{t}
= \sum_{k=1}^\infty rac{(-1)^{k+1}x^k}{k\; k!}

.

Using this function, we then may define, using the logarithm,

:

{
m E}_1(x) \,=\, -\gamma-\ln x + {
m Ein}(x)

and

:

{
m Ei}(x) \,=\, \gamma+\ln x - {
m Ein}(-x).

The exponential integral may also be generalized to

:

E_n(x) = \int_1^\infty rac{e^{-xt}}{t^n}\, dt

which is sometimes called Misra function

arphi_m(x)\,

, defined as

:

arphi_m(x)=E_{-m}(x)\,

REFERENCES

Milton Abramowitz and Irene A. Stegun, eds. '' Handbook Of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.'' New York: Dover, 1972. '' (See Chapter 5) ''

http://mathworld.wolfram.com/En-Function.html

R. D. Misra, Proc. Cambridge Phil. Soc. 36, 173 (1940)

	CATEGORIES ABOUT EXPONENTIAL INTEGRAL

	exponentials

	special functions

	special hypergeometric functions

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Information About

Exponential Integral