Introduction to Bessel Functions

CHAPTER I

BESSEL FUNCTIONS OF ZERO ORDER

§ 1. Bessel’s function of zero order.

The function known as Bessel’s function of zero order, and denoted by J0(x), may be defined by the infinite power-series

(1.1)

If ur denotes the rth term of this series, we have

which → 0 when r → ∞, whatever the value of x. Consequently, the series converges for all values of x, and since it is a power-series, the function J0(x) and all its derivatives are continuous for all values of x, real or complex.

§ 2. Bessel’s function of order n, when n is a positive integer.

The function Jn(x), known as Bessel’s function of order n, may be defined, when n is a positive integer, by the infinite power-series

(1.2)

which converges for all values of x, real or complex.

In particular, when n = 1 we have

(1.3)

and when n = 2

(1.4)

We note that Jn(x) is an even function of x when n is even, odd when n is odd.

The graphs of J0(x), J1(x) are indicated in Fig. 1.

FIG. 1.

Extensive tables of values of Jn(x), especially of J0(x) and J1(x), have been calculated on account of their applications to physical problems.¹

§ 3. Bessel’s equation of zero order.

By differentiating the series for J0(x) and comparing the result with the series for J1(x), we find ²

(1.5)

Again, after multiplying the series for J1(x) by x and differentiating, we find

(1.6)

Using (1.5), we can write (1.6) in the form

(1.7)

or

(1.8)

Thus y = J0(x) satisfies the linear differential equation of the second order

(1.9)

or

(1.10)

or

(1.11)

which is known as Bessel’s equation of zero order.

§ 4. Bessel functions of the second kind of zero order.

A solution of Bessel’s equation which is not a numerical multiple of J0(x) is called a Bessel function of the second kind. Let u be such a function, and let v = J0(x); then, by (1.10),

Multiplying the first of these equations by v and the second by u and subtracting, we have

x(u′′v − uv′′) + u′v − uv′ = 0,

which, since

can be written

Hence

x(u′v − uv′) = B,

where B is a constant. Dividing by xv², we have

that is,

and hence, by integration,

Consequently, since v = J0(x),

. (1.12)

where A, B are constants, and B ≠ 0 since u is not a constant multiple of J0(x), by definition.

§ 5. If, in the last integral, J0(x) is replaced by its series, and the integrand expanded in ascending powers of x, we find for the first few terms

and therefore

Consequently, if we put

(1.13)

then Y0(x) is a particular Bessel function of the second kind; it is called Neumann’s Bessel function of the second kind of zero order; the general term in its expansion can be obtained by other methods (§ 8).

Since J0(x) → 1 when x → 0, it follows from (1.13) that Y0(x) behaves like log x when x is small, and hence that Y0(x) → − ∞ when x → + 0.

§ 6. It follows from (1.12) that every Bessel function of the second kind of zero order can be written in the form

AJ0(x) + BY0(x).

The one that has been most extensively tabulated is Weber’s,³ which is denoted by Y0(x) and is obtained by putting

and hence

(1.14)

where γ denotes Euler’s constant, defined by

(1.15)

We note that, when x is small,

(1.16)

the remaining terms being small in comparison with unity.

As far as applications are concerned, it is usually sufficient to bear in mind that Y0(x) is a Bessel function of the second kind whose values have been tabulated; that x must be positive for Y0(x) to be real, on account of the term involving log x in (1.13); and that Y0(x) → − ∞ when x → + 0.

The graphs of J0(x) and Y0(x) are shown together in Fig. 2.

§ 7. General solution of Bessel’s equation of zero order.

Since J0(x) and Y0(x) are independent solutions of the equation

the general solution can be written

(1.17)

where A, B are arbitrary constants, and x > 0 for Y0(x) to be real.

If we replace x by kx, where k is a constant, the equation becomes

FIG. 2.

Multiplying by k², we deduce that the general solution of the equation

(1.18)

can be written

(1.19)

where k > 0 for Y0(kx) to be real when x > 0.

§ 8. The general solution by Frobenius’s method.

Bessel’s equation belongs to the type to which Frobenius’s method of solution in series can be applied. Put

(1.20)

and make the substitution

(1.21)

We obtain, after collecting like terms,

(1.22)

Now let c1, c2, c3, . . . be chosen to satisfy the equations

(ρ + 1)²c1 = 0,

(ρ + 2)² c2 + 1 = 0,

(ρ + 3)² c3 + c1 = 0, . . .

Then, unless ρ is a negative integer,

Substituting these values in (1.21) and (1.22), we deduce that, if

(1.23)

and if ρ is not a negative integer, then

(1.24)

Putting p = 0 in (1.23) and (1.24) we see again that

is a solution of Bessel’s equation

Further, differentiating (1.24) partially with respect to p, we get

and hence, when p = 0,

from which it follows that (∂y/∂ρ)ρ=0 is a second solution. Now, from (1.23),

Hence, putting p = 0 and Y0(x) = (∂y/∂ρ)ρ=0 we obtain the second solution

(1.25)

which is Neumann’s Bessel function of the second kind of zero order, in a form which indicates the general term (§ 5).

It follows that the general solution of the equation can be written

y = AJ0(x) + BY0(x),

which is equivalent to (1.17).

§ 9. To examine the convergence of the series that follows J0(x) log x in (1.25), we can put, by (1.15),

(1.26)

where ∊n → 0 when n → ∞. Hence if ur denote the rth term of the series, we have

which → 0 when r → ∞, whatever the value of x. Consequently, the series converges absolutely for all values of x, real or complex.

§ 10. Integrals.

We notice next certain integrals involving Bessel functions in their integrands. Firstly, from (1.5) and (1.6) we have

(1.27)

(1.28)

Secondly, we note that the indefinite integral

(1.29)

cannot be expressed in a simpler form, but on account of its importance the value of the definite integral

(1.30)

has been tabulated.⁴

Thirdly, we shall obtain a reduction formula for the integral

(1.31)

Put

un = ʃxnJ0(x)dx = ʃxn−1d{xJ1(x)},

by (1.6). Then, integrating by parts, we have

by (1.5); and on integrating by parts again,

un = xnJ1(x) + (n − 1)xn−1J0(x) − (n − 1)²ʃxn−2J0(x)dx,

that is,

(1.32)

which is the reduction formula required.

It follows that, if n is a positive integer,