BESSEL'S EQUATION AND BESSEL FUNCTIONS

Bessel Equation and Bessel Function:

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number α, the order of the Bessel function.
        The differential equation

     x²y" + xy' + (x² - p²)y = 0 .......(1)

  Where p is a real constant is called BESSEL'S(1784 _ 1846)EQUATION of order p.

          It is clear that x=0 is a regular singular point of the equation . Hence we assume a solution of the form
         ∞
   y = Σ   cₙxⁿ⁺ʳ                           .........(2)
        n=0
where c₀≠0

Substitution of the series for y , y'  and y" in equation (1) yields
     ∞                                     ∞
     Σ (n+r)(n+r-1) cₙxⁿ⁺ʳ + Σ (n+r)cₙxⁿ⁺ʳ
    n=0                                 n=0
         ∞                      ∞
      + Σ cₙxⁿ⁺ʳ⁺² - p² Σ cₙxⁿ⁺ʳ =0
        n=0                  n=0
         ∞
or     Σ {(n+r)(n+r-1) +(n+r) - p²}cₙxⁿ⁺ʳ
       n=0
                         ∞
                      + Σ  cₙ₋₂xⁿ⁺ʳ = 0
                       n=2
            ∞                 ∞
(since Σ cₙxⁿ⁺ʳ⁺² = Σ cₘ₋₂xᵐ⁺ʳ and
          n=0              m=2

m is dummy index )

or   (r² - p²)c₀xʳ + {(r+1)² - p²}c₁xʳ⁺¹
            ∞
         + Σ [{(n+r)² - p²}cₙ + cₙ₋₂]xⁿ⁺ʳ = 0 ........(3)
           n=0

Equating to zero the co _ efficient of lowest power of x (i.e., xʳ) in equation (3) we obtain the indicial equation r² - p² =0 , which has the roots r₁ = p and r₂ = -p . Equating to zero co_ efficients of higher powers of x  in equation (3) , we obtain

         {(r+1)² - p²}c₁ = 0

and  {(r+n)² - p² }cₙ + cₙ₋₂ = 0 , n≥2

which imply c₁ = 0

              cₙ = - cₙ₋₂ / (r+n)² - p²

Hence   c₂ = - c₀ / (r+2)² - p² ; c₃ = 0

               c₄ = - c₂/ (r+4)² - p²

                   = c₀ / {(r+4)² - p²}{(r+2)² - p²} ; c₅=0

    c₆ = - c₄ / (r+6)² - p²

      = - c₀ / {(r+6)² - p²}{(r+4)² - p²}{(r+2)² - p²}

Therefore ,from equation (2)

y = c₀xʳ[1 - x² / (r+2)² - p²

            + x⁴ / {(r+2)² - p²}{(r+4)² - p²}

  - x⁶ / {(r+2)² - p²}{(r+4)² - p²}{(r+6)² - p²} + ..]
                                                 ...........(4)

    = c₀ W (say)


Which is required solution of Bessel Differential Equation.
If p is not an integer , then the two roots p and -p of the indicial equation yield two linearly independent solutions y₁(x) and y₂(x) given by

y₁(x) = c₀W]ᵣ₌ ₚ = c₀xᵖ[1 - x²/ 2(2p+2)

                                      + x⁴ / 2(2p+2)4(2p+4)

                     - x⁶ / 2(2p+2)(2p+4)4(2p+6)6 + ....]

          = c₀xᵖ [1 - x²/ 2²(p+1)

                        + x⁴ / 2⁴ 2! (p+1)(p+2)

                 - x⁶ / 2⁶3! (p+1)(p+2)(p+3) + ....]

y₂(x) = c₀W]ᵣ₌ ₋ₚ = c₀x⁻ᵖ[1 - x² / 2²(1-p)

                              + x⁴ /2! 2⁴ (1-p)(2-p)

                       - x⁶ / 2⁶ 3! (1-p)(2-p)(3-p) +...]

To standardise the solution ,we take

           c₀ = 1/ 2ᵖ Γ (p+1)

and denote the first solution y₁(x) by Jₚ(x).

Thus , Jₚ(x) = xᵖ / 2ᵖ Γ(p+1) [ 1- x² / 2²(p+1)

                                    + x⁴ / 2! 2⁴(p+1)(p+2)

                         - x⁶ / 3! 2⁶(p+1) (p+2)(p+3)+ ...]

     = 1 / Γ(p+1)(x/2)ᵖ - 1/ 1! Γ(p+2)(x/2)ᵖ⁺²

 + 1/ 2! Γ(p+3)(x/2)ᵖ⁺⁴ - 1/ 3! Γ(p+4)(x/2)ᵖ⁺⁶+...
                  ∞
i.e. Jₚ(x) = Σ (-1)ⁿ / n! Γ(p+n+1) (x/2)ᵖ⁺²ⁿ  .....(5)
                 n=0

Jₚ(x) is defined as the Bessel function .

Graphically Bessel Function can be shown as follows;

See Also:

More about BESSEL FUNCTIONS

About Scientist:

 Friedrich Bessel. Friedrich Wilhelm Bessel (German: [ˈbɛsəl]; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax.

For More