3.1. Bessel’s Equation:
As we have pointed above that the equation
…………(1)
In which is a constant, is known as Bessel’s equation and its solutions are called as Bessel functions. These functions were used by Bessel in the nineteenth century in a problem of dynamical Astronomy but since an orthonormal set of function can be defined with Bessel functions it has enormous use in different branches of mathematical physics and Engineering. We have shown the two solutions of (1) may be taken as and [ is not an integer] when
……….…….(2)
If is an integer then
…………..(3)
Now, using and hence is infinite when r = 0, 1, 2….(n-1). Therefore
………….(4)
Writing r = s + n, this become
……….(5)
In this case the other solution is taken to Yn(x) [given in (8) in 1.2]
Example. show that
Putting in (2)
For the second relation put in (3)
The results are special cases of an important general theorem which states that is expressible in finite terms by means of algebraic and trigonometric function of x whenever is half of an odd integer. The functions and when n is a positive integer or zero are called spherical Bessel functions and have important application in wave motion in which spherical polar coordinates are appropriate.
2. Bessel function of order zero
When the first solution of Bessel’s equation is given by
(1)
Let Yo (x) is the second independent solution and we take it as
[e.f. 1.2]
………….(2)
when Y is the Euler’s constant and is a series is ascending powers of x, Now
Substituting in Bessel equation with i.e.
………….(3)
We find
i.e. ………(4)
To solve (4) let
i.e.
i.e.
comparing with (4) we get
taking the above recurrence relation gives and
these gives successively
and in general
substitution in (2) we get
……….(5)
Yo(x) is called the Bessel function of the second kind of order zero and the complete solution of Bessel’s equation of order zero (3) is
…………….(6)
Both Jo(x) and Yo(x) are oscillatory functions and their graphs are shown below. It should be noted that both Jo(x) and Yo(x) vanish at an infinite sequence of values of x. In this respect they behave similarly to the trigonometrical functions cos x and sin x which vanish respectively when and . In practical application the positive values of x for which the Bessel function vanish are of great importance and these values are known as positive zeroes of the functions. If we denote the rth zero of Jo(x), Yo(x) by and respectively, It can be shown that these zeroes interlace i.e. for all r. Unlike those of the trigonometrical functions, the positive zeroes of the Bessel function are not equally spaced along the x-axis, but it can be shown that and approximate respectively to and for large values of r. Numerical values of the zeroes can be found in many hand book and numerical Tables. It is easily seen that for small values of x
Thus and also it can be shown that both the function tends to zero as
Note: can also be developed as above. In fact from (6) and (8) of 1.2 we can write
……….(6)
the first term on the right being x/2. Below we give the rough graphs of J1(x) and Y1(x). Note the interlacing of their positive zeroes. For small x, J1(x) and Y1(x) behaves like x/2 and respectively. Both the function tends to zero as. These positive zeroes are extensively tabulated. To avoid the necessity of having to write the general solution of Bessel’s equation in the two different forms.
i.e. not zero or positive integer
n zero or a positive integer. It is possible to define the second solution as
………….(7)
and then for (n zero or integer) the expression on r,h,s, of (7) tends to Yn(x) [Use L’ Hospital’s theorem]. Thus with the form (7), the general solution of Bessel’s equation of order for all values of is ………(8)
3. Hankel functions.
Although and are independent solutions of Bessel’s equation it is sometimes convenient to take the fundamental solution in a slightly different form. Henkel functions often called Bessel functions of the third kind, are often defined by
………..(9)
And there are independent solutions of Bessel’s equation. In terms of these function, the general solution of the equation is
………..(10)
The Hankel functions bear the same relation to the Bessel function of the first and third kind as the function bear to and and they are convenient in mathematical analysis for similar reasons. Note that by (7) and (9)
………(11)
Use of in the propagation of waves with and if the time dependent part is generates forward wave and generates backward wave.
4. Some properties of Bessel functions.
Bessel function possesses properties of which great use can be made in the discussion of physical problems. A few of the more important of these are given in this section but for an extensive treatment one may refer to the standard text by Watson.
(a)Recurrence formulae
We have seen that
i.e.
……..(1)
This can be written in the form
And it follows that
………….(2)
This is the first of the required recurrence formulae. A useful particular case of this is
Again,
We have
……(3)
This can be written
i.e.
…………(4)
Two other useful recurrence formulae followed by addition and subtraction of (2) and (4). Thus
.…….…(5)
………..(6)
The function of the second kind and the Hankel functions satisfy the same recurrence formulae. For example
Thus adding and using (5), we have
The other recurrence formulae can be proved in the same way.
(b) The generating function for the Bessel coefficients: Provided t is not zero, the functions and can be expanded in powers of t, and product of these expansion gives
Term by term multiplication of the series being permissible because of the absolute convergence of the separate series. The coefficients of tn(n a positive integer or zero) is found by taking n = r-s i.e. r = n+s and taking s very form 0 to infinity. Thus the coefficients of tn is
The coefficients of t -n (n a positive integer) is found by taking r = -n+s and taking s very from n to infinity. Thus the coefficient of t -n = [see equ. (4) of 3.1]
Hence ……….(8)
And the exponential function on the left can be regarded as the generating function of Jn(x). Because of the form of (8), the functions Jn(x) when n=0, 1, 2, 3… are often called the Bessel coefficients.
Example: Show that
Writing
Since J-n(x) =(-1)n Jn(x). The required result follows when we note that and
(c) Bessel Integral
Equation (9) can be written as
………(10)
Since when r=0 and vanishes when r is an integer, multiplication of (10) by and integration w.r.t. between o and 2 gives formally
………(11)
n = 0, 1, 2, 3,
A rigorous proof however exists else where [Lebedeo –Special function and their applications] since Jn(x) is real, we can show by equating real and imaginary parts that
and the expression (11) gives Bessel’s integral for Jn(x).
(d)Some integrals involving Bessel functions.
Using the result (3)
If is real and positive then as and so
………(13)
and a similar integral can do obraiaed from the result (1). Other integrals of this type can be evaluated by such devices as integration by parts and use of recurrence formulae. Below we discuss an important integral
……….(14)
Let then u satisfies Bessel’s equation when u is written in place of y and in place of x. This can be written in the form,
………(15)
Similarly satisfies the equation
……..(16)
Multiplying by and respectively and subtracting
….(17)
Substituting for u,v and integrating w.r.t. x between o and a
And we have obtained the value of I in (14) when i.e.
If the integral (14) becomes
and from (18)
[L’ Hospital’s theorem]
……….(19)
Since satisfies the Bessel’s equation
=0 ………….....(20)
We get using (19) and (20)
…….(21)
Interesting results comes out when is such that
This case we shall discuss below.
(e) Fourier Bessel series
Taking when and are two positive roots of the equation and substituting in (18) and (21) we have
…………………(22)
…………….(23)
Hence the functions where are the roots of =0 are orthogonal in the interval (0, a), and the possibility of expanding an arbitry functions in the form
……(24)
Assuming this to be possible [for a certain class of function ]
multiplication of (24) by and the use of (22) and (23) leads to
…………………(25)
the other term on the r.h.s. vanishes by the virtue of (22). The recurrence relation (2) gives
and since it follows that
Thus (25) gives
……………………………(26)
Equation (26) gives a formulae from which the coefficient in the series on right of (24) can be calculated and, from the analogy with the Fourier series, this series is called Fourier Bessel series.
The analysis given above is, of course, purely formal and no attempt has been made to discuss the condition under which such an expansion of an arbitrary function is legitimate. A full discussion of these will be found in Watson’s standard treaties. By choosing to be the roots of equations other than = 0, other expansions can be obtained, an example will be found in Exercise III [usually needed in Acoustics or Electro-Magnetic diffraction in a mixed boundary value problem].
Asymptotic Values
In physical problems it is often describe to be able to approximate to the Bessel functions and when x is large For such values of , the series defining these function converge slowly but useful approximation can be obtained as follows. Writing in Bessel’s equation we obtain
…………(1)
and we seek a solution of the form substituting in (1), the term with the highest power of is and equating its coefficient to zero, with this value of the recurrence relation between the coefficients and is found in the usual way to be
……………..(2)
The resulting series is not convergent but it can, nevertheless, be used to find numerical values of the solution. For large values of we can take a first approximation to the solution of Bessel’s equation to be i.e. and another approximate solution can similarly be found to be . The standard solution can be identified with combinations of these approximate solutions by writing
…………………………….(3)
It is easy to verify that when , these approximation yield the exact solution . The result (3) can of course be improved by using the recurrence relation (2) to calculate the coefficients further turns in the series giving in descending powers of . Such series are called asymptotic series and the reader is referred to more advanced treaties for full discussion.
Modified Bessel function.
Writing in place of , Bessel’s equation becomes
………………(1)
And this ‘modified’ equation plays a significant role in science and engineering. Working exactly in the same way as in example 2, one solution of this equation is given by the function
……………..(2)
The introduction of in the second part of Equation (2) is useful but it is of course, only formal as we have not defined the Bessel function for imaginary arguments. However this procedure can be justified by appealing to works discussing Bessel function of the complex variable. The function is called the modified Bessel function of the first kind of order and it is a solution of Bessel’s modified equation for all values of .
As was the case with Bessel’s equation, certain difficulties arise in finding a second solution of the modified equation when (). It is conventional to denote the second solution by and to define it by the relation.
……………(3)
or by the limit of the expression on the right The function so defined is known as the modified Bessel function of the second kind of order and it can be shown that
+ ………..(4)
While for
+
……(5)
The term corresponding to in the second series being
In terms of these functions, the general solution of the modified equation (1) is, for all values of
………………………………(6)
Where A, B are arbitrary constants.
The modified function which occur most frequently in practical applications are those of order zero and unity. Rough graph are shown
It is more convenient to plot than the function themselves because of the behavior of the function for large value of . This can be shown to be
, ……………(7)
The modified function , bear to exponential function .
Similar relations to those which the function bear to the trigonometrical function, and the modified functions have no zeroes for real values of . Some useful properties are given in the exercises. Numerical values for various may be found in Tables.