TOPICS
Search

Lambda Function


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

LambdaFunction

The triangle function, illustrated above, is commonly denoted Lambda(x).

LambdaFunctionJahnke

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

 Lambda_nu(z)=Gamma(nu+1)(J_nu(z))/((1/2z)^nu)
(1)

where J_n(z) is a Bessel function of the first kind and Gamma(x) is the gamma function. Lambda_0(z)=J_0(z), and taking nu=1 gives the special case

 Lambda_1(z)=(J_1(z))/(1/2z)=2jinc(z),
(2)

where jinc(z) is the jinc function.

A two-variable lambda function is defined as

 lambda(x,y)=int_0^y(Gamma(t+1)dt)/(x^t),
(3)

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


See also

Airy Functions, Carmichael Function, Dirichlet Lambda Function, Elliptic Lambda Function, Jinc Function, Liouville Function, Mangoldt Function, Mu Function, Nu Function, Triangle Function

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Gradshteyn, I. S. and Ryzhik, I. M. "The Functions nu(x), nu(x,a), mu(x,beta), mu(x,beta,alpha), lambda(x,y)." §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.

Referenced on Wolfram|Alpha

Lambda Function

Cite this as:

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

Subject classifications