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Lambda Function


The lambda function defined by Jahnke and Emden (1945) is

$\displaystyle \Lambda_\nu(z)$ $\textstyle \equiv$ $\displaystyle \Gamma(\nu+1) {J_\nu(z)\over ({\textstyle{1\over 2}}z)^\nu}$ (1)
$\displaystyle \Lambda_1(z)$ $\textstyle \equiv$ $\displaystyle {J_1(z)\over {z\over 2}}=2\mathop{\rm jinc}\nolimits (z),$ (2)

where $J_1(z)$ is a Bessel Function of the First Kind and $\mathop{\rm jinc}\nolimits (z)$ is the Jinc Function.

A two-variable lambda function defined by Gradshteyn and Ryzhik (1979) is

\lambda(x,y)\equiv \int_0^y {\Gamma(u+1)\,du\over x^u},
\end{displaymath} (3)

where $\Gamma(z)$ is the Gamma Function.

See also Airy Functions, Dirichlet Lambda Function, Elliptic Lambda Function, Jinc Function, Lambda Hypergeometric Function, Mangoldt Function, Mu Function, Nu Function


Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1079, 1979.

Jahnke, E. and Emde, F. Tables of Functions with Formulae and Curves, 4th ed. New York: Dover, 1945.

© 1996-9 Eric W. Weisstein