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Bourget Function

The function defined by the contour integral

J_(n,k)(z) =1/(2pii)int^((0+))t^(-n-1)(t+1/t)^kexp[1/2z(t-1/t)]dt,

where int_((0+)) denotes the contour encircling the point z=0 once in a counterclockwise direction. It is equal to

J_(n,k)(z)=1/piint_0^pi(2costheta)^kcos(ntheta-zsintheta)dtheta

(Watson 1966, p. 326).

See also

Bessel Function of the First Kind

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References

Bourget, J. "Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste." J. de Math. 6, 33-54, 1861.Giuliani, G. "Alcune osservazioni sopra le funzioni spheriche di ordine superiore al secondo e sopra altre funzioni che se ne possono dedurre (April, 1888)." Giornale di Mat. 26, 155-171, 1888.Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, p. 465, 1988.Watson, G. N. "The Functions of Bourget and Giuliani." §10.31 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 326-327, 1966.

Referenced on Wolfram|Alpha

Bourget Function

Cite this as:

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

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