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E_t-Function

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A function which arises in the fractional integral of e^(at), given by

E_t(nu,a)=(e^(at))/(Gamma(nu))int_0^tx^(nu-1)e^(-ax)dx
(1)
=(a^(-nu)e^(at)gamma(nu,at))/(Gamma(nu)),
(2)

where gamma(a,z) is the incomplete gamma function and Gamma(z) the complete gamma function.

The E_t function satisfies the recurrence relation

E_t(nu,a)=aE_t(nu+1,a)+(t^nu)/(Gamma(nu+1)).
(3)

A special value is

E_t(0,a)=lim_(nu->0)E_t(nu,a)=e^(at).
(4)

See also

En-Function, Fractional Calculus, Fractional Integral

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Cite this as:

Weisstein, Eric W. "E_t-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Et-Function.html

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