Lambda Function

There are a number of functions in mathematics commonly denoted with a Greek letter lambda. Examples of one-variable functions denoted with a lower case lambda include the Carmichael functions, Dirichlet lambda function, elliptic lambda function, and Liouville function. Examples of one-variable functions denoted with an upper case lambda include the Mangoldt function and the lambda function defined by Jahnke and Emden (1945).

The triangle function, illustrated above, is commonly denoted .

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

(1)

where is a Bessel function of the first kind and is the gamma function. , and taking gives the special case

(2)

where is the jinc function.

A two-variable lambda function is defined as

(3)

where is the gamma function (McLachlan et al. 1950, p. 9; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109).