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 equation which 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.

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