Et-Function

A function which arises in Fractional Calculus.

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

where $\gamma$ is the incomplete Gamma Function and $\Gamma$ the complete Gamma Function. The $E_t$ function satisfies the Recurrence Relation

$$E_t(\nu,a)=aE_t(\nu+1,a)+{t^\nu\over\Gamma(\nu+1)}.$$ (2)

A special value is

$$E_t(0,a)=e^{at}.$$ (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.