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Bessel functions of the first and second kind. Bessel’s differential equation. Hankel functions. Modified Bessel functions. Recurrence formulas.

Bessel’s differential equation. The equation

1)        x²y" + xy' + (x² - ν²)y = 0

where ν is real and 0 is known as Bessel’s equation of order ν. Solutions of this equation are called Bessel functions of order ν.

Bessel functions of the first kind. The function

is known as the Bessel function of the first kind of order ν. The formula is valid providing ν -1, -2, -3, .... . Γ(ν) is the gamma function.

The Bessel function

is obtained by replacing ν in 2) with a -ν.

● If ν = 0, 1, 2, 3, .... , J_-ν(x) = (-1)^ν J_ν(x)

● If ν 0, 1, 2, 3, .... , J_ν(x) and J_-ν(x) are linearly independent.

● If ν 0, 1, 2, 3, .... , J_ν(x) is bounded at x = 0 while J_-ν(x) is unbounded.

For ν = 0, 1

The graphs of J₀(x) and J₁(x) are shown in Fig. 1. One notes their similarity to the graphs of sin x and cos x. These graphs illustrate the important fact that the equation J_ν(x) = 0 has infinitely many roots for every value of ν.

Generating function for J_n(x). For n a positive or negative integer, the n-th Bessel function, J_n(x), is the coefficient of tⁿ in the expansion of

 in powers of t and 1/t i.e.

Bessel functions of the second kind. Bessel functions Y_ν(x) of the second kind are defined as follows:

Case 1. ν is a non-integer.

Case 2. ν is a positive integer n = 0, 1, 2, 3, ....

where γ = 0.5772156.... is Euler’s constant and

For n = 0,

Case 3. ν is a negative integer 0, -1, -2, -3, .....

Fig. 2 shows Y₀(x) and Y_1.(x)

● For any value, ν 0, J_ν(x) is bounded at x = 0 while Y_ν(x) is unbounded.

Bessel functions of the third kind. Hankel functions. Bessel functions of the third kind are also called Hankel functions. Hankel functions of the first and second kinds are defined as

respectively.

General solution of Bessel’s Differential Equation. The general solution of Bessel’s differential equation is given by any of the following:           

1)        y = A J_ν(x) + B J-_ν(x)                      (valid for ν a non-integer)

2)        y = A J_ν(x) + BY_ν(x)                       (valid for all values of ν)

where A and B are arbitrary constants.

Recurrence formulas for the Bessel functions

● The functions Y_ν(x) satisfy identical relations.

Bessel functions of order equal to odd multiples of one half. Bessel functions of order equal to n·(1/2) where n = 1, 3, 5, .... can be expressed in terms of sines and cosines.

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Bessel’s modified differential equation. The equation

1)        x²y" + xy' - (x² + ν²)y = 0

where ν is real and 0 is known as Bessel’s modified differential equation of order ν. Solutions of this equation are called modified Bessel functions of order ν.

Modified Bessel functions of the first kind. The function

is known as the modified Bessel function of the first kind of order ν. The formula is valid providing ν -1, -2, -3, ....

The modified Bessel function

is obtained by replacing ν in 2) with a -ν.

● If ν = 0, 1, 2, 3, .... I_-ν(x) = I_ν(x)

● If ν 0, 1, 2, 3, .... I_ν(x) = I_-ν(x) are linearly independent.

For ν = 0, 1

Generating function for I_n(x). For ν = n, an integer, the function

is a generating function for I_n(x), that is,

Modified Bessel functions of the second kind. Modified Bessel functions K_ν(x) of the second kind are defined as follows:

Case 1. ν is a non-integer.

Case 2. ν is a positive integer n = 0, 1, 2, 3, ....

where γ = 0.5772156.... is Euler’s constant and

For n = 0,

Case 3. ν is a negative integer 0, -1, -2, -3, .....

General solution of Bessel’s Modified Differential Equation. The general solution of Bessel’s modified differential equation is given by any of the following:  

1)        y = A I_ν(x) + B I-_ν(x)           (valid for ν a non-integer)

2)        y = A I_ν(x) + B K_ν(x)           (valid for all values of ν)

where A and B are arbitrary constants.

Recurrence formulas for modified Bessel functions

First kind

Second kind

Modified Bessel functions of order equal to odd multiples of one half. Modified Bessel functions of order equal to n·(1/2) where n = 1, 3, 5, .... can be expressed in terms of sines and cosines.

Integral representations for Bessel functions

Asymptotic expansions

Orthogonal series of Bessel functions. Let λ₁, λ₂, λ₃, ..... be the positive roots of

            R J_ν(x) + S x J_ν(x) = 0                    ν > -1

Then the following series expansions hold under the conditions indicated.

Case 1. S = 0, R 0, i.e. λ₁, λ₂, λ₃, ..... are the positive roots of J_ν(x) = 0.

            f(x) = A₁ J_ν(λ₁x) + A₂ J_ν(λ₂x) + A₃ J_ν(λ₃x) + .......

where 

In particular, if n = 0

            f(x) = A₁ J₀(λ₁x) + A₂ J₀(λ₂x) + A₃ J₀(λ₃x) + .......

where

Case 2. R/S > -ν

            f(x) = A₁ J_ν(λ₁x) + A₂ J_ν(λ₂x) + A₃ J_ν(λ₃x) + .......

where 

In particular, if n = 0

            f(x) = A₁ J₀(λ₁x) + A₂ J₀(λ₂x) + A₃ J₀(λ₃x) + .......

where

Case 3. R/S = -ν

            f(x) = A₁ J_ν(λ₁x) + A₂ J_ν(λ₂x) + A₃ J_ν(λ₃x) + .......

where

In particular, if n = 0

            f(x) = A₁ J₀(λ₁x) + A₂ J₀(λ₂x) + A₃ J₀(λ₃x) + .......

where

Case 4. R/S < -ν. In this case there are two pure imaginary roots iλ₀ in addition to the positive roots λ₁, λ₂, λ₃, ..... . We have

            f(x) = A I_ν(λ₀x) + A₁ J_ν(λ₁x) + A₂ J_ν(λ₂x) + A₃ J_ν(λ₃x) + .......

where

Miscellaneous formulas

1)        cos (x sin θ) = J₀(x) + 2 J₂(x) cos 2θ + 2 J₄(x) cos 4θ + ........

2)        sin (x sin θ) = 2 J₁(x) sin θ + 2 J₃(x) cos 3θ + 2 J₅(x) cos 5θ + ........

            (called the addition formula for Bessel functions)

4)        1 = J₀(x) + 2 J₂(x) + ..... + 2 J_2n(x) + .........

5)        x = 2 [J₁(x) + 3 J₃(x) + 5 J₅(x) + ..... + (2n + 1) J_2n+1(x) + ......... ]

6)        x² = 2 [4 J₂(x) + 16 J₄(x) + 36 J₆(x) + ..... + (2n²) J_2n(x) + ......... ]

            Formulas 9) and 10) can be generalized.

14)      sin x = 2 [J₁(x) - J₃(x) + J₅(x) - ........ ]

15)      cos x = J₀(x) - 2 J₂(x) + 2 J₄(x) - ........

16)      sinh x = 2 [I₁(x) + I₃(x) + I₅(x) + ........ ]

15)      cosh x = I₀(x) + 2 [I₂(x) + I₄(x) + I₆(x) + ........ ]

Indefinite integrals

Definite integrals

References.

1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

2. International Dictionary of Applied Mathematics

3. Wylie. Advanced Engineering Mathematics

4. James / James. Mathematics Dictionary

Website owner:  James Miller