Bessel Function of the Second Kind

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A Bessel function of the second kind 👁 Y_n(x)
(e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted 👁 N_n(x)
(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 👁 Y_n(x)
for 👁 n=0
, 1, 2, ..., 5. The Bessel function of the second kind is implemented in the Wolfram Language as [nu, z].

Let 👁 v=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×
(1) minus 👁 u×
(2),

👁 x(u^('')v-uv^(''))+u^'v-uv^'=0

(3)

👁 d/(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^')/(v^2)=d/(dx)(u/v)=B/(xv^2)

(5)

👁 u/v=A+Bint(dx)/(xv^2).

(6)

Rearranging and using 👁 v=J_m(x)
gives

👁 u	👁 =	👁 AJ_m(x)+BJ_m(x)int(dx)/(xJ_m^2(x))	(7)
👁 Image	👁 =	👁 A^'J_m(x)+B^'Y_m(x),	(8)

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

👁 Y_nu(z)
can be defined by

👁 Y_nu(z)=(J_nu(z)cos(nupi)-J_(-nu)(z))/(sin(nupi))

(9)

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

👁 Y_n(z)=-((1/2z)^(-n))/pisum_(k=0)^(n-1)((n-k-1)!)/(k!)(1/4z^2)^k+2/piln(1/2z)J_n(z)-((1/2z)^n)/pisum_(k=0)^infty[psi_0(k+1)+psi_0(n+k+1)]((-1/4z^2)^k)/(k!(n+k)!),

(10)

where 👁 psi_0(x)
is the digamma function (Abramowitz and Stegun 1972, p. 360).

The function has the integral representations

👁 Y_nu(z)	👁 =	👁 1/piint_0^pisin(zsintheta-nutheta)dtheta-1/piint_0^infty[e^(nut)+e^(-nut)(-1)^nu]e^(-zsinht)dt	(11)
👁 Image	👁 =	👁 -(2(1/2z)^(-nu))/(sqrt(pi)Gamma(1/2-nu))int_1^infty(cos(zt)dt)/((t^2-1)^(nu+1/2))	(12)

(Abramowitz and Stegun 1972, p. 360).

Asymptotic series are

👁 Y_m(x)	👁 ∼	👁 {2/pi[ln(1/2x)+gamma] m=0,x<<1; -(Gamma(m))/pi(2/x)^m m!=0,x<<1	(13)
👁 Y_m(x)	👁 ∼	👁 sqrt(2/(pix))sin(x-(mpi)/2-pi/4) x>>1,	(14)

where 👁 Gamma(z)
is a gamma function.

👁 BesselY0ReIm

👁 BesselY0Contours

For the special case 👁 n=0
, 👁 Y_0(x)
is given by the series

👁 Y_0(z)=2/pi{[ln(1/2z)+gamma]J_0(z)+sum_(k=1)^infty(-1)^(k+1)H_k((1/4z^2)^k)/((k!)^2)},

(15)

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

Related Wolfram sites

http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/

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References

Abramowitz, M. and Stegun, I. 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.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 👁 Y_nu(x)
." 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.

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Bessel Function of the Second Kind

Cite this as:

Weisstein, Eric W. "Bessel Function of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

URL: https://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html