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).

REFERENCES:

Gradshteyn, I. S. and Ryzhik, I. M. "The Functions , , , , ." §9.64 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1109, 2000.

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

McLachlan, N. W. et al. Supplément au formulaire pour le calcul symbolique. Paris: L'Acad. des Sciences de Paris, Fasc. 113, p. 9, 1950.

Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990.