The linear differential equation

x2 + x + (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 = xr ∑ k=0∞a kxk = ∑ k=0∞a kxr+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):

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

(3)

The last of those can be written

Because a₀≠0, the first of those (the indicial equation) 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)a_k + a_k−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

(4)

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

y1 := a0xp

(5)

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

or r₂ + k≠p = r₁. Therefore

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

y2 := a0x−p

(6)

The power series of (5) and (6) converge for all values of x and are linearly independent (the ratio y₁∕y₂ tends to 0 as x →∞). With the appointed value

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

where J_−p(x) = y₂ with a₀ = .

The explicit expressions for J_±p are

J±p(x) = ∑ m=0∞2m±p,

(7)

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

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

Thus we get

analogically (6) yields

and the general solution of the equation (1) for p = 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)ⁿJ _n(x), i.e. linearly dependent of J_n(x). It can be shown that the other solution of (1) ought to be searched in the form y = K_n(x) = J_n(x) ln x + x⁻ⁿ ∑ _k=0^∞b _kx^k. Then the general solution is y := C ₁J_n(x) + C₂K_n(x).

Other formulae

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

F(z, t) = e (t−) = ∑ n=−∞∞J n(z)tn

(8)

This function has an essential singularity at t = 0 but is analytic elsewhere in ℂ; 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) := e ^(t−), ζ := gives

The paths γ and δ go once round the origin anticlockwise in the t-plane and ζ-plane, respectively. Since the residue of ζ^−m−n−1e^ζ in the origin is = , the residue theorem gives

This means that F has the Laurent expansion (8).

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

Also one can obtain the addition formula

and the series representations of cosine and sine:

References