Bessel’s equation

The linear differential equation

x2d2ydx2+xdydx+(x2-p2)y= 0, (1)

in which p is a constant (non-negative if it is real), is called the Bessel’s equation.  We derive its general solution by trying the series form

y=xrk=0akxk=k=0akxr+k, (2)

due to Frobenius.  Since the parameter r is indefinite, we may regard a0 as distinct from 0.

We substitute (2) and the derivatives of the series in (1):

x2k=0(r+k)(r+k-1)akxr+k-2+xk=0(r+k)akxr+k-1+(x2-p2)k=0akxr+k= 0.

Thus the coefficients of the powers xr, xr+1, xr+2 and so on must vanish, and we get the system of equations

{[r2-p2]a0= 0,[(r+1)2-p2]a1= 0,[(r+2)2-p2]a2+a0= 0,[(r+k)2-p2]ak+ak-2= 0. (3)

The last of those can be written

(r+k-p)(r+k+p)ak+ak-2= 0.

Because  a00,  the first of those (the indicial equationMathworldPlanetmath) gives  r2-p2=0,  i.e. we have the roots


Let’s first look the the solution of (1) with  r=p;  then  k(2p+k)ak+ak-2=0,  and thus


From the system (3) we can solve one by one each of the coefficients a1, a2,   and express them with a0 which remains arbitrary.  Setting for k the integer values we get

{a1= 0,a3= 0,,a2m-1= 0;a2=-a02(2p+2),a4=a024(2p+2)(2p+4),,a2m=(-1)ma0246(2m)(2p+2)(2p+4)(2p+2m) (4)

(where  m=1, 2,). Putting the obtained coefficients to (2) we get the particular solution

y1:=a0xp[x22(2p+2)+x424(2p+2)(2p+4)-x6246(2p+2)(2p+4)(2p+6)+-] (5)

In order to get the coefficients ak for the second root  r2=-p  we have to look after that

(r2+k)2-p2 0,

or  r2+kp=r1.  Therefore

r1-r2= 2pk

where k is a positive integer.  Thus, when p is not an integer and not an integer added by 12, we get the second particular solution, gotten of (5) by replacing p by -p:

y2:=a0x-p[1-x22(-2p+2)+x424(-2p+2)(-2p+4)-x6246(-2p+2)(-2p+4)(-2p+6)+-] (6)

The power seriesMathworldPlanetmath of (5) and (6) converge for all values of x and are linearly independentMathworldPlanetmath (the ratio y1/y2 tends to 0 as  x).  With the appointed value


the solution y1 is called the Bessel functionDlmfMathworldPlanetmathPlanetmath of the first kind and of order p and denoted by Jp.  The similar definition is set for the first kind Bessel function of an arbitrary order  p (and ). For  p  the general solution of the Bessel’s differential equation is thus


where  J-p(x)=y2  with  a0=12-pΓ(-p+1).

The explicit expressions for J±p are

J±p(x)=m=0(-1)mm!Γ(m±p+1)(x2)2m±p, (7)

which are obtained from (5) and (6) by using the last for gamma functionDlmfDlmfMathworldPlanetmath.

E.g. when  p=12  the series in (5) gets the form


Thus we get


analogically (6) yields


and the general solution of the equation (1) for  p=12  is


In the case that p is a non-negative integer n, the “+” case of (7) gives the solution


but for  p=-n  the expression of J-n(x) is (-1)nJn(x), i.e. linearly dependent on Jn(x).  It can be shown that the other solution of (1) ought to be searched in the form  y=Kn(x)=Jn(x)lnx+x-nk=0bkxk.  Then the general solution is  y:=C1Jn(x)+C2Kn(x).

Other formulae

The first kind Bessel functions of integer order have the generating function F:

F(z,t)=ez2(t-1t)=n=-Jn(z)tn (8)

This functionMathworldPlanetmath has an essential singularity at  t=0  but is analytic elsewhere in ; thus F has the Laurent expansion in that point.  Let us prove (8) by using the general expression


of the coefficients of Laurent series.  Setting to this  a:=0,  f(t):=ez2(t-1t),  ζ:=zt2  gives


The paths γ and δ go once round the origin anticlockwise in the t-plane and ζ-plane, respectively.  Since the residueDlmfPlanetmath of ζ-m-n-1eζ in the origin is  1(m+n)!=1Γ(m+n+1),  the residue theoremMathworldPlanetmath ( gives


This that F has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g. the of the Bessel functions of integer order:


Also one can obtain the addition formulaPlanetmathPlanetmath


and the series of cosine and sine:

sinz= 2J1(z)-2J3(z)+2J5(z)-+


  • 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  Kirjastus Valgus, Tallinn (1966).
  • 2 K. Kurki-Suonio: Matemaattiset apuneuvot.  Limes r.y., Helsinki (1966).
Title Bessel’s equation
Canonical name BesselsEquation
Date of creation 2013-03-22 16:34:57
Last modified on 2013-03-22 16:34:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Definition
Classification msc 34A05
Classification msc 33C10
Synonym Bessel’s differential equation
Synonym Bessel equation
Related topic LaplaceEquationInCylindricalCoordinates
Related topic CauchyResidueTheorem
Related topic PropertiesOfEntireFunctions
Related topic FrobeniusMethod
Related topic TableOfLaplaceTransforms
Related topic BesselFunctionsAndHelicalStructureDiffractionPatterns
Defines Bessel’s function
Defines Bessel function