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Special functions. Bessel’s equation of order ν. Bessel function of the first kind.

There are a number of functions in mathematics that have come to be referred to as special functions. For example: Gamma function, Beta function, Bessel functions, Legendre functions; Hermite, Laguerre and Chebyshev polynomials. Where did these functions come from? Most arose in connection with the series solution of important differential equations. When a particular differential equation arises frequently enough, is important enough, it receives its own name. When a particular series arises frequently enough, is important enough, it receives its own name.

We will now illustrate how both the gamma function and the Bessel function arise in connection with the series solution of the Bessel differential equation.

One of the most important of all variable-coefficient differential equations is

1)        x²y" + xy' + (λ²x² - ν²)y = 0

which is known as Bessel’s equation of order ν with parameter λ. This equation arises in a great variety of problems in physics and engineering including almost all partial differential equation problems having circular symmetry as in the wave or heat equation. And it arises in many applications where neither circular symmetry nor partial differential equations are involved.

As a first step in solving this equation we do a change of variables from x to t by means of the substitution

2)        t = λx.

Now  

3)        dt = λdx

 and dividing 3) by dy and inverting we get

From 4) we get

Substitution of 2), 4) and 5) into 1) then gives

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

We have thus just shown that the problem of solving Bessel’s equation of order ν with parameter λ can be reduced to the problem of solving Bessel’s equation of order ν by employing a change of variable. We shall now derive the solution of Bessel’s equation of order ν.

Problem. Solve Bessel’s equation of order ν

7)        x²y" + xy' + (x² - ν²)y = 0

in the vicinity of x = 0 where ν is real and 0 and we shall assume that it is not an integer.

Solution. This equation has a regular singular point at x = 0.

Step 1. Assume solution. Assume a solution of the form

8)        y = x^c[a₀ + a₁x + a₂x² + a₃x³ + ....... ] = a₀x^c + a₁x^c+1 + a₂x^c+2 + a₃x^c+3 + .......

where a₀ 0.

Step 2. Substitute assumed solution into differential equation. Substitute the assumed series 8) into the differential equation 7), collect like terms, and do an index shift in such a way as to bring all the exponents of x down to the smallest one present.

Substitution of 8) into 7) gives

which reduces to

A shift in index, replacing n with n - 2 in the second term, gives 

Step 3. Compute roots of indicial equation. The equations for determining c and the a’s are

10)      n = 0:  (c² - ν²)a₀ = 0

            n = 1: [(1 + c)² - ν²]a₁ = 0

            n 2:      [(n + c)² - ν²)a_n + a_{n -2} = 0

The indicial equation is then given by

            (c² - ν²) = 0

 Its roots are c₁ = -ν and c₂ = ν .

Computation of solution corresponding to the first root.

Step 4. Derive recurrence relation for first root. We will do the root c = ν first. Substituting c = ν into equations of 10) we get

11)      0· a₀ = 0       (a₀ arbitrary)

            a₁ = 0

            n(n + 2ν)a_n + a_n-2 = 0           n 2

which gives the recurrence relation

Step 5. Set up column and use multiplication devise to obtain a_n. Since a₁ = 0, all terms with odd subscripts will be zero. Substituting into 6) for terms with even subscripts we get, for different values of n,

            a₀ arbitrary

            -------------

Multiplying these equations together and simplifying we get

or

This represents the expression for the coefficients. Let us make some modifications for purposes of simplification. Let us multiply both numerator and denominator by the gamma function Γ(ν+1). This gives

The denominator now contains the product

            Γ(ν +1)·(ν +1)(ν +2)(ν +3) ...... (ν + k) .

This product is equal to Γ(ν + k +1).

Proof

Equation 14) then becomes

For reasons that will be soon apparent let us now multiply the numerator and denominator of 14) by 2^ν to obtain

The constant a₀ is arbitrary and since we are only looking for a particular solution let us assign a₀ the value

giving

The solution can then be written as

which is the Bessel function of the first kind of order ν. Since Bessel’s equation has no finite singular points except the origin the series converges for all values of x if v 0. The usual notation for the Bessel function of the first kind of order ν is J_ν(x). Thus 17) can be written as

We could now derive the second solution to our differential equation by going back and substituting the second root c = -ν of the indicial equation into the recurrence relation 12) and repeating steps 4 through 6. We will not, however, do this. We will simply state the result. It is

Note that y₂ can be obtained from y₁ simply by replacing ν by -ν in the series.

J_ν(x) and J-_ν(x) constitute two linearly independent solutions of Bessel’s equation. The complete solution of Bessel’s equation is

            y(x) = c₁ J_ν(x) + c₂ J_-ν(x)

where c₁ and c₂ are arbitrary constants.

References

1. Earl Rainville. Elementary Differential Equations.

2. Wylie. Advanced Engineering Mathematics.