Bessel Function of the Second Kind

A Bessel function of the second kind $Y_n(x)$ 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 $Y_n(x)$ for $n=1$, 2, ..., 5.

Let $v\equiv J_m(x)$ be the first solution and $u$ be the other one (since the Bessel Differential Equation is second-order, there are two Linearly Independent solutions). Then

$$xu''+u'+xu=0$$ (1)

$$xv''+v'+xv=0.$$ (2)

Take $v\times$ (1) minus $u\times$ (2),

$$x(u''v-uv'')+u'v-uv'=0$$ (3)

$${d\over dx} [x(u'v-uv')]= 0,$$ (4)

so $x(u'v-uv')=B$, where $B$ is a constant. Divide by $xv^2$,

$${u'v-uv'\over v^2} = {d\over dx} \left({u\over v}\right)= {B\over xv^2}$$ (5)

$${u\over v}=A+B\int{dx\over xv^2}.$$ (6)

Rearranging and using $v\equiv J_m(x)$ gives

$$u = AJ_m(x)+BJ_m(x)\int {dx\over x{J_m}^2(x)}$$

$$\equiv A'J_m(x)+B'Y_m(x),$$ (7)

where the Bessel function of the second kind is defined by

$Y_m(x) = {J_m(x)\cos(m\pi)-J_{-m}(x)\over\sin(m\pi)}$
$= {1\over\pi} \sum_{k=1}^\infty {(-1)^kx^{m+2k}\over 2^{m+2k}k!(m+k)!} \left[{2\ln\left({x\over 2}\right)+2\gamma-b_{m+k}-b_k}\right]$
$- {1\over\pi} \sum_{k=0}^{m-1} {x^{-m+2k}(m-k-1)!\over 2^{-m+2k}k!}\quad$	(8)

$m = 0$, 1, 2, ..., $\gamma$ is the Euler-Mascheroni Constant, and

$$b_k \equiv \cases{ 0 & $k = 0$,\cr \sum_{n=0}^k {1\over n} & $k \not = 0$.\cr}$$ (9)

The function is given by

$$Y_n(z) = {1\over\pi}\int_0^\pi \sin(z\sin\theta-n\theta)\,d\... ...ver\pi} \int_0^\infty [e^{nt}+e^{-nt}(-1)^n]e^{-z\sinh t}\,dt.$$ (10)

Asymptotic equations are

$$Y_m(x) = \left\{\begin{array}{ll} {2\over\pi}\left[{\ln({\textstyle{1\over... ... \pi}\left({2\over x}\right)^m & \mbox{$m \not = 0, x \ll 1$}\end{array}\right.$$ (11)

$$Y_m(x) = \sqrt{2\over\pi x} \,\sin\left({x-{m\pi\over 2}-{\pi\over 4}}\right)\quad x \gg 1,$$ (12)

where $\Gamma(z)$ is a Gamma Function.

See also Bessel Function of the First Kind, Bourget's Hypothesis, Hankel Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Bessel Functions $J$ and $Y$.'' §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, $N_\nu(x)$.'' §11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596-604, 1985.

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 $Y_\nu(x)$.'' Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533-542, 1987.