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Et-Function

A function which arises in Fractional Calculus.

\begin{displaymath}
E_t(\nu,a)={1\over\Gamma(\nu)} e^{at} \int_0^t x^{\nu-1}e^{-ax}\,dx=t^\nu e^{at} \gamma(\nu,at),
\end{displaymath} (1)

where $\gamma$ is the incomplete Gamma Function and $\Gamma$ the complete Gamma Function. The $E_t$ function satisfies the Recurrence Relation
\begin{displaymath}
E_t(\nu,a)=aE_t(\nu+1,a)+{t^\nu\over\Gamma(\nu+1)}.
\end{displaymath} (2)

A special value is
\begin{displaymath}
E_t(0,a)=e^{at}.
\end{displaymath} (3)

See also En-Function


References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Exponential Integral and Related Functions.'' Ch. 5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 227-233, 1972.




© 1996-9 Eric W. Weisstein
1999-05-25