Bessel & Legendre function

24 BESSEL FUNCTIONS

BESSEL'S DIFFERENTIAL EQUATION

24.1

x²y'' + xy' + (x² - n²)y = 0   n ≥ 0

Solutions of this equation are called Bessel functions of order n.

BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n

24.2

J_n(x) = xⁿ / 2ⁿΓ(n+1) { 1 - x² / 2(2n+2) + x⁴ / 2 · 4(2n+2)(2n+4) - ... }

= ∑_k=0^∞ (-1)^k(x/2)^n+2k / k! Γ(n+k+1)

24.3

J_-n(x) = x^-n / 2^-nΓ(1-n) { 1 - x² / 2(2-2n) + x⁴ / 2 · 4(2-2n)(4-2n) - ... }

= ∑_k=0^∞ (-1)^k(x/2)^2k-n / k! Γ(k+1-n)

24.4

J_-n(x) = (-1)ⁿJ_n(x)   n = 0, 1, 2, ...

If n ≠ 0, 1, 2, ..., J_n(x) and J_-n(x) are linearly independent.

If n ≠ 0, 1, 2, ..., J_n(x) is bounded at x = 0 while J_-n(x) is unbounded.

For n = 0, 1 we have

24.5

J₀(x) = 1 - x² / 2² + x⁴ / 2² · 4² - x⁶ / 2² · 4² · 6² + ...

24.6

J₁(x) = x / 2 - x³ / 2² · 4 + x⁵ / 2² · 4² · 6 - x⁷ / 2² · 4² · 6² · 8 + ...

24.7

J₀'(x) = -J₁(x)

BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n

24.8

Y_n(x) = { J_n(x) cos nπ - J_-n(x) ∕ sin nπ   n ≠ 0, 1, 2, ...

lim_p→n J_p(x) cos pπ - J_-p(x) ∕ sin pπ   n = 0, 1, 2, ... }

This is also called Weber's function or Neumann's function [also denoted by N_n(x)].

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BESSEL FUNCTIONS

BESSEL'S DIFFERENTIAL EQUATION

24.1

x²y'' + xy' + (x² - n²)y = 0   n ≧ 0

Solutions of this equation are called Bessel functions of order n.

BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n

24.2

J_n(x) = ^xn/_{2ⁿΓ(n + 1)} { 1 - ^x2/_{2(2n + 2)} + ^x4/_{2 · 4(2n + 2)(2n + 4)} - ... }

     =  Σ^∞_{k = 0} ^{(-1)k(x/2)n + 2k}/_{k! Γ(n + k + 1)}

24.3

J_-n(x) = x^-nJ_n(x) { 1 - ^x2/_{2(2 - 2n)} + ^x4/_{2 · 4(2 - 2n)(4 - 2n)} + ... }

     =  Σ^∞_{k = 0} ^{(-1)k(x/2)2k-n}/_{k! Γ(k + 1 - n)}

24.4

J_-n(x) = (-1)ⁿJ_n(x)   n = 0, 1, 2, ...

If n ≠ 0, 1, 2, ..., J_n(x) and J_-n(x) are linearly independent.

If n ≠ 0, 1, 2, ..., J_n(x) is bounded at x = 0 while J_-n(x) is unbounded.

For n = 0, 1 we have

24.5

J₀(x) = 1 - ^x2/_2² + ^x4/_{2² · 4²} - ^x6/_{2² · 4² · 6²} + ...

24.6

J₁(x) = ^x/₂ - ^x3/_{2² · 4} + ^x5/_{2² · 4² · 6} - ^x7/_{2² · 4² · 6² · 8} + ...

24.7

J₀'(x) = -J₁(x)

BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n

24.8

Y_n(x) = {^{J_n(x) cos nπ - J_-n(x)}/_{sin nπ}   n ≠ 0, 1, 2, ...

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