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Mu Function

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There are two functions commonly denoted mu, each of which is defined in terms of integrals. Another unrelated mathematical function represented using the Greek letter mu is the Möbius function.

MuFunction

The two-argument mu-function is defined by the definite integral

mu(x,beta)=int_0^infty(x^tt^betadt)/(Gamma(beta+1)Gamma(t+1)),

where Gamma(z) is the gamma function (Erdélyi et al. 1981a, p. 388; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109), while the three-argument mu-function is defined by

mu(x,beta,alpha)=int_0^infty(x^(alpha+t)t^betadt)/(Gamma(beta+1)Gamma(alpha+t+1))

(Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109).

See also

Lambda Function, Nu Function

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 388, 1981a.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Ch. 18 in Higher Transcendental Functions, Vol. 3. New York: Krieger, p. 217, 1981b.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.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990.

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Mu Function

Cite this as:

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

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