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Beta Exponential Function

BetaExp

Another "beta function" defined in terms of an integral is the "exponential" beta function, given by

beta_n(z)=int_(-1)^1t^ne^(-zt)dt
(1)
=n!z^(-(n+1))[e^zsum_(k=0)^(n)((-1)^kz^k)/(k!)-e^(-z)sum_(k=0)^(n)(z^k)/(k!)].
(2)

If n is an integer, then

beta_n(z)=(-1)^(n+1)E_(-n)(-z)-E_(-n)(z),
(3)

where E_n(z) is the En-Function. The exponential beta function satisfies the recurrence relation

zbeta_n(z)=(-1)^ne^z-e^(-z)+nbeta_(n-1)(z).
(4)

The values for n=0, 1, and 2 are

beta_0(z)=(2sinhz)/z
(5)
beta_1(z)=(2(sinhz-zcoshz))/(z^2)
(6)
beta_2(z)=(2(2+z^2)sinhz-4zcoshz)/(z^3).
(7)

See also

Alpha Function, En-Function

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Cite this as:

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

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