Integral exponential function

Presentation of the functions ${\displaystyle \operatorname {E_{1}} (x)}$

Presentation of the functions ${\displaystyle \operatorname {Ei} (x)}$

Presentation of the functions ${\displaystyle \operatorname {li} (x)}$

Presentation of the functions ${\displaystyle \operatorname {Ein} (x)}$

In mathematics , the integral exponential function is called ${\displaystyle \operatorname {Ei} (x)}$

${\displaystyle \operatorname {Ei} (x)=\int _{-\infty }^{x}{\frac {e^{t}}{t}}\,\mathrm {d} t=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,\mathrm {d} t}$

Are defined.

Since at diverges, the above integral for is to be understood as Cauchy's principal value . ${\displaystyle {\tfrac {1}{t}}}$ ${\displaystyle t=0}$ ${\displaystyle x>0}$

The integral exponential function has the series representation

${\displaystyle \operatorname {Ei} (x)=\gamma +\ln \left|x\right|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}\ ,}$

where is the natural logarithm and the Euler-Mascheroni constant . ${\displaystyle \ln }$ ${\displaystyle \gamma }$

The integral exponential function is closely related to the integral logarithm , it holds ${\displaystyle \operatorname {li} (x)}$

${\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln x)\quad 0<x\neq 1.}$

Also closely related is a function that integrates via another integration area:

${\displaystyle \operatorname {E} _{1}(x)=\int _{1}^{\infty }{\frac {e^{-tx}}{t}}\,\mathrm {d} t=\int _{x}^{\infty }{\frac {e^{-t}}{t}}\,\mathrm {d} t.}$

This function can be understood as an extension of the integral exponential function to negative real values, since

${\displaystyle \operatorname {Ei} (-x)=-\operatorname {E} _{1}(x).}$

With the help of the whole function

${\displaystyle \operatorname {Ein} (x)=\int _{0}^{x}{\frac {1-e^{-t}}{t}}\,\mathrm {d} t=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}x^{k}}{k!k}}}$

can be the other two as

${\displaystyle \operatorname {E} _{1}(x)=-\gamma -\ln x+\operatorname {Ein} (x)}$

or.

${\displaystyle \operatorname {Ei} (x)=\gamma +\ln x-\operatorname {Ein} (-x)}$

represent.

The integral exponential function is a special case of the incomplete gamma function

${\displaystyle E_{n}(x)=x^{n-1}\Gamma (1-n,x).}$

It can also be used as a

${\displaystyle E_{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}}{t^{n}}}\,\mathrm {d} t\quad \Re (x)>0}$

to be generalized.

literature

• William H. Press et al .: Numerical Recipes (FORTRAN) . Cambridge University Press, New York 1989.
• Milton Abramowitz , Irene A. Stegun (Eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Dover, New York 1972. (See Chapter 5) .
• RD Misra: Proc. Cambridge Phil. Soc. Volume 36, 1940, p. 173 (Please check! Doubtful according to JFM, strange title: On the stability of crystal lattices. II , p. 173–182)

Web links

• Eric W. Weisstein : Exponential Integral . In: MathWorld (English).
• Eric W. Weisstein : En-Function . In: MathWorld (English).
• Maxim Lwowitsch Konzewitsch : Exponential Integral . Lecture series (English), 2015.