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Beta Function (Exponential)


Another ``Beta Function'' defined in terms of an integral is the ``exponential'' beta function, given by

$\displaystyle \beta_n(z)$ $\textstyle \equiv$ $\displaystyle \int_{-1}^1 t^ne^{-zt}\,dt$ (1)
  $\textstyle =$ $\displaystyle n!z^{-(n+1)}\left[{e^z\sum_{k=0}^n {(-1)^kz^k\over k!} -e^{-z}\sum_{k=0}^n {z^k\over k!}}\right].$  

The exponential beta function satisfies the Recurrence Relation
\end{displaymath} (3)

The first few integral values are
$\displaystyle \beta_0(z)$ $\textstyle =$ $\displaystyle {2\sinh z\over z}$ (4)
$\displaystyle \beta_1(z)$ $\textstyle =$ $\displaystyle {2(\sinh z-z\cosh z)\over z^2}$ (5)
$\displaystyle \beta_2(a)$ $\textstyle =$ $\displaystyle {2(2+z^2)\sinh z-4z\cosh z\over z^3}.$ (6)

See also Alpha Function

© 1996-9 Eric W. Weisstein