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Find a General Solution in Terms of Bessel Functions


Bessel Function of the First Kind

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BesselJ

The Bessel functions of the first kind J_n(x) are defined as the solutions to the Bessel differential equation

x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0

(1)

which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows J_n(x) for n=0, 1, 2, ..., 5. The notation J_(z,n) was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written J_n(2z) (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function

e^(z(t-1/t)/2)=sum_(n=-infty)^inftyt^nJ_n(z).

(2)

is the same as the modern one (Watson 1966, p. 14). Bessel used the notation I_k^h to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).

The Bessel function J_n(z) can also be defined by the contour integral

J_n(z)=1/(2pii)∮e^((z/2)(t-1/t))t^(-n-1)dt,

(3)

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ[nu, z].

To solve the differential equation, apply Frobenius method using a series solution of the form

y=x^ksum_(n=0)^inftya_nx^n=sum_(n=0)^inftya_nx^(n+k).

(4)

Plugging into (1) yields

x^2sum_(n=0)^infty(k+n)(k+n-1)a_nx^(k+n-2)+xsum_(n=0)^infty(k+n)a_nx^(k+n-1)+x^2sum_(n=0)^inftya_nx^(k+n)-m^2sum_(n=0)^inftya_nx^(n+k)=0

(5)

sum_(n=0)^infty(k+n)(k+n-1)a_nx^(k+n)+sum_(n=0)^infty(k+n)a_nx^(k+n) +sum_(n=2)^inftya_(n-2)x^(k+n)-m^2sum_(n=0)^inftya_nx^(n+k)=0.

(6)

The indicial equation, obtained by setting n=0, is

a_0[k(k-1)+k-m^2]=a_0(k^2-m^2)=0.

(7)

Since a_0 is defined as the first nonzero term, k^2-m^2=0, so k=+/-m. Now, if k=m,

sum_(n=0)^infty[(m+n)(m+n-1)+(m+n)-m^2]a_nx^(m+n)+sum_(n=2)^inftya_(n-2)x^(m+n)=0

(8)

sum_(n=0)^infty[(m+n)^2-m^2]a_nx^(m+n)+sum_(n=2)^inftya_(n-2)x^(m+n)=0

(9)

sum_(n=0)^inftyn(2m+n)a_nx^(m+n)+sum_(n=2)^inftya_(n-2)x^(m+n)=0

(10)

a_1(2m+1)x^(m+1)+sum_(n=2)^infty[a_nn(2m+n)+a_(n-2)]x^(m+n)=0.

(11)

First, look at the special case m=-1/2, then (11) becomes

sum_(n=2)^infty[a_nn(n-1)+a_(n-2)]x^(m+n)=0,

(12)

so

a_n=-1/(n(n-1))a_(n-2).

(13)

Now let n=2l, where l=1, 2, ....

a_(2l) = -1/(2l(2l-1))a_(2l-2)

(14)

= ((-1)^l)/([2l(2l-1)][2(l-1)(2l-3)]...[2·1·1])a_0

(15)

= ((-1)^l)/(2^ll!(2l-1)!!)a_0,

(16)

which, using the identity 2^ll!(2l-1)!!=(2l)!, gives

a_(2l)=((-1)^l)/((2l)!)a_0.

(17)

Similarly, letting n=2l+1,

a_(2l+1)=-1/((2l+1)(2l))a_(2l-1)=((-1)^l)/([2l(2l+1)][2(l-1)(2l-1)]...[2·1·3][1])a_1,

(18)

which, using the identity 2^ll!(2l+1)!!=(2l+1)!, gives

a_(2l+1)=((-1)^l)/(2^ll!(2l+1)!!)a_1=((-1)^l)/((2l+1)!)a_1.

(19)

Plugging back into (◇) with k=m=-1/2 gives

y = x^(-1/2)sum_(n=0)^(infty)a_nx^n

(20)

= x^(-1/2)[sum_(n=1,3,5,...)^(infty)a_nx^n+sum_(n=0,2,4,...)^(infty)a_nx^n]

(21)

= x^(-1/2)[sum_(l=0)^(infty)a_(2l)x^(2l)+sum_(l=0)^(infty)a_(2l+1)x^(2l+1)]

(22)

= x^(-1/2)[a_0sum_(l=0)^(infty)((-1)^l)/((2l)!)x^(2l)+a_1sum_(l=0)^(infty)((-1)^l)/((2l+1)!)x^(2l+1)]

(23)

= x^(-1/2)(a_0cosx+a_1sinx).

(24)

The Bessel functions of order +/-1/2 are therefore defined as

so the general solution for m=+/-1/2 is

y=a_0^'J_(-1/2)(x)+a_1^'J_(1/2)(x).

(27)

Now, consider a general m!=-1/2. Equation (◇) requires

a_1(2m+1)=0

(28)

[a_nn(2m+n)+a_(n-2)]x^(m+n)=0

(29)

for n=2, 3, ..., so

for n=2, 3, .... Let n=2l+1, where l=1, 2, ..., then

where f(n,m) is the function of l and m obtained by iterating the recursion relationship down to a_1. Now let n=2l, where l=1, 2, ..., so

a_(2l) = -1/(2l(2m+2l))a_(2l-2)

(34)

= -1/(4l(m+l))a_(2l-2)

(35)

= ((-1)^l)/([4l(m+l)][4(l-1)(m+l-1)]...[4·(m+1)])a_0.

(36)

Plugging back into (◇),

y = sum_(n=0)^(infty)a_nx^(n+m)=sum_(n=1,3,5,...)^(infty)a_nx^(n+m)+sum_(n=0,2,4,...)^(infty)a_nx^(n+m)

(37)

= sum_(l=0)^(infty)a_(2l+1)x^(2l+m+1)+sum_(l=0)^(infty)a_(2l)x^(2l+m)

(38)

= a_0sum_(l=0)^(infty)((-1)^l)/([4l(m+l)][4(l-1)(m+l-1)]...[4(m+1)])x^(2l+m)

(39)

= a_0sum_(l=0)^(infty)([(-1)^lm(m-1)...1]x^(2l+m))/([4l(m+l)][4(l-1)(m+l-1)]...[4(m+1)m...1])

(40)

= a_0sum_(l=0)^(infty)((-1)^lm!)/(2^(2l)l!(m+l)!)x^(2l+m).

(41)

Now define

J_m(x)=sum_(l=0)^infty((-1)^l)/(2^(2l+m)l!(m+l)!)x^(2l+m),

(42)

where the factorials can be generalized to gamma functions for nonintegral m. The above equation then becomes

y=a_02^mm!J_m(x)=a_0^'J_m(x).

(43)

Returning to equation (◇) and examining the case k=-m,

a_1(1-2m)+sum_(n=2)^infty[a_nn(n-2m)+a_(n-2)]x^(n-m)=0.

(44)

However, the sign of m is arbitrary, so the solutions must be the same for +m and -m. We are therefore free to replace -m with -|m|, so

a_1(1+2|m|)+sum_(n=2)^infty[a_nn(n+2|m|)+a_(n-2)]x^(|m|+n)=0,

(45)

and we obtain the same solutions as before, but with m replaced by |m|.

J_m(x)={sum_(l=0)^(infty)((-1)^l)/(2^(2l+|m|)l!(|m|+l)!)x^(2l+|m|) for |m|!=1/2; sqrt(2/(pix))cosx for m=-1/2; sqrt(2/(pix))sinx for m=1/2.

(46)

We can relate J_m(x) and J_(-m)(x) (when m is an integer) by writing

J_(-m)(x)=sum_(l=0)^infty((-1)^l)/(2^(2l-m)l!(l-m)!)x^(2l-m).

(47)

Now let l=l^'+m. Then

But l^'!=infty for l^'=-m,...,-1, so the denominator is infinite and the terms on the left are zero. We therefore have

Note that the Bessel differential equation is second-order, so there must be two linearly independent solutions. We have found both only for |m|=1/2. For a general nonintegral order, the independent solutions are J_m and J_(-m). When m is an integer, the general (real) solution is of the form

Z_m=C_1J_m(x)+C_2Y_m(x),

(52)

where J_m is a Bessel function of the first kind, Y_m (a.k.a. N_m) is the Bessel function of the second kind (a.k.a. Neumann function or Weber function), and C_1 and C_2 are constants. Complex solutions are given by the Hankel functions (a.k.a. Bessel functions of the third kind).

The Bessel functions are orthogonal in [0,a] according to

int_0^aJ_nu(alpha_(num)rho/a)J_nu(alpha_(nun)rho/a)rhodrho=1/2a^2[J_(nu+1)(alpha_(num))]^2delta_(mn),

(53)

where alpha_(num) is the mth zero of Jnu and delta_(mn) is the Kronecker delta (Arfken 1985, p. 592).

Except when 2m is a negative integer,

J_m(z)=(z^(-1/2))/(2^(2m+1/2)i^(m+1/2)Gamma(m+1))M_(0,m)(2iz),

(54)

where Gamma(x) is the gamma function and M_(0,m) is a Whittaker function.

In terms of a confluent hypergeometric function of the first kind, the Bessel function is written

J_nu(z)=((1/2z)^nu)/(Gamma(nu+1))_0F_1(nu+1;-1/4z^2).

(55)

A derivative identity for expressing higher order Bessel functions in terms of J_0(z) is

J_n(z)=i^nT_n(id/(dz))J_0(z),

(56)

where T_n(z) is a Chebyshev polynomial of the first kind. Asymptotic forms for the Bessel functions are

J_m(z) approx 1/(Gamma(m+1))(z/2)^m

(57)

for z<<1 and

J_m(z) approx sqrt(2/(piz))cos(z-(mpi)/2-pi/4)

(58)

for z>>|m^2-1/4| (correcting the condition of Abramowitz and Stegun 1972, p. 364).

A derivative identity is

d/(dx)[x^mJ_m(x)]=x^mJ_(m-1)(x).

(59)

An integral identity is

int_0^uu^'J_0(u^')du^'=uJ_1(u).

(60)

Some sum identities are

sum_(k=-infty)^inftyJ_k(x)=1

(61)

(which follows from the generating function (◇) with t=1),

1=[J_0(x)]^2+2sum_(k=1)^infty[J_k(x)]^2

(62)

(Abramowitz and Stegun 1972, p. 363),

1=J_0(x)+2sum_(k=1)^inftyJ_(2k)(x)

(63)

(Abramowitz and Stegun 1972, p. 361),

0=sum_(k=0)^(2n)(-1)^kJ_k(z)J_(2n-k)(z)+2sum_(k=1)^inftyJ_k(z)J_(2n+k)(z)

(64)

for n>=1 (Abramowitz and Stegun 1972, p. 361),

J_n(2z)=sum_(k=0)^nJ_k(z)J_(n-k)(z)+2sum_(k=1)^infty(-1)^kJ_k(z)J_(n+k)(z)

(65)

(Abramowitz and Stegun 1972, p. 361), and the Jacobi-Anger expansion

e^(izcostheta)=sum_(n=-infty)^inftyi^nJ_n(z)e^(intheta),

(66)

which can also be written

e^(izcostheta)=J_0(z)+2sum_(n=1)^inftyi^nJ_n(z)cos(ntheta).

(67)

The Bessel function addition theorem states

J_n(y+z)=sum_(m=-infty)^inftyJ_m(y)J_(n-m)(z).

(68)

Various integrals can be expressed in terms of Bessel functions

J_n(z)=1/piint_0^picos(zsintheta-ntheta)dtheta,

(69)

which is Bessel's first integral,

for n=1, 2, ...,

J_n(z)=2/pi(z^n)/((2n-1)!!)int_0^(pi/2)sin^(2n)ucos(zcosu)du

(72)

for n=1, 2, ...,

J_n(x)=1/(2pii)int_gammae^((x/2)(z-1/z))z^(-n-1)dz

(73)

for n>-1/2. The Bessel functions are normalized so that

int_0^inftyJ_n(x)dx=1

(74)

for positive integral (and real) n. Integrals involving J_1(x) include

int_0^infty[(J_1(x))/x]^2dx=4/(3pi)

(75)

int_0^infty[(J_1(x))/x]^2xdx=1/2.

(76)

Ratios of Bessel functions of the first kind have continued fraction

(J_(n-1)(z))/(J_n(z))=(2n)/z-(z/(2(n+1)))/(1-(((z/2)^2)/((n+1)(n+2)))/((1-((z/2)^2)/((n+2)(n+3)))/(1-...)))

(77)

(Wall 1948, p. 349).

BesselJ0ReIm BesselJ0Contours

The special case of n=0 gives J_0(z) as the series

J_0(z)=sum_(k=0)^infty(-1)^k((1/4z^2)^k)/((k!)^2)

(78)

(Abramowitz and Stegun 1972, p. 360), or the integral

J_0(z)=1/piint_0^pie^(izcostheta)dtheta.

(79)

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Find a General Solution in Terms of Bessel Functions

Source: https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html