Bessel Function of the Second Kind

A Bessel function of the second kind  (e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted  (e.g, Gradshteyn and Ryzhik 2000, p. 657, eqn. 6.518), is a solution to the Bessel differential equationwhich is singular at the origin. Bessel functions of the second kind are also called Neumann functions or Weber functions. The above plot shows  for , 1, 2, ..., 5. The Bessel function of the second kind is implemented in the Wolfram Language as BesselY[nu, z].

Let  be the first solution and  be the other one (since the Bessel differential equation is second-order, there are two linearly independent solutions). Then

			(1)
			(2)

Take  (1) minus  (2),

(3)

(4)

so , where  is a constant. Divide by ,

(5)

(6)

Rearranging and using  gives

			(7)
			(8)

where  is the so-called Bessel function of the second kind.

 can be defined by

(9)

(Abramowitz and Stegun 1972, p. 358), where  is a Bessel function of the first kind and, for  an integer  by the series

(10)

where  is the digamma function (Abramowitz and Stegun 1972, p. 360).

The function has the integral representations

			(11)
			(12)

(Abramowitz and Stegun 1972, p. 360).

Asymptotic series are

			(13)
			(14)

where  is a gamma function.

For the special case ,  is given by the series

(15)

(Abramowitz and Stegun 1972, p. 360), where  is the Euler-Mascheroni constant and  is a harmonic number.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions  and ." §9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972.

Arfken, G. "Neumann Functions, Bessel Functions of the Second Kind, ." §11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596-604, 1985.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 625-627, 1953.

Spanier, J. and Oldham, K. B. "The Neumann Function ." Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533-542, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.