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Bessel Function of the Second Kind

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BesselY

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 BesselY[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)
=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)
=-(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.

See also

Bessel Function of the First Kind, Bourget's Hypothesis, Hankel Function, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind

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 Web Resource. https://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

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