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) = ^(x/2)n/(n!) [ 1 − ^x2/_{2(2n + 2)} + ^x4/_{2 · 4(2n + 2)(2n + 4)} − ... ]

= ^{∑ (−1)k(x/2)2k}/_{k! Γ(n + k + 1)}

24.3

J_−n(x) = ^(x/2)n/(−n)! [ 1 − ^x2/_{2(2 − 2n)} + ^x4/_{2 · 4(2 − 2n)(4 − 2n)} − ... ]

= ^{∑ (−1)k(x/2)2k−n}/_{k!(−n + k)}

24.4

J_−n(x) = (−1)ⁿ J_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/_2²4² + ...

24.6

J₁(x) = ^x/₂ − ^x3/_{2 · 2²} + ^x5/_{2 · 2 · 6²} + ...

24.7

J₀'(x) = −J₁(x)

BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n

24.8

Y_n(x) = ^{∫ J_n(xt) cos xπ − J_n(x)}/_{sin xπ} n ≠ 0, 1, 2, ...

= ^{lim J_n(x) cos pπ − J_−n(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

For x = 0, 1, 2, ..., L'Hospital's rule yields

24.9

Y_n(x) = ₂/^{π(ln (x/2) + γ)y_n(x) - π}Σ_k-0^n-1((n-k-1)!)_k! ((x/2)^2k-n

- ₁/^π∑_k=0^∞(-1)^k(x

/2)^2k+n : k!(n+k)!

where γ =.5772156... is Euler's constant

24.10

Φ(0) = 1 - ₁/² 1 Φ(0) = 1

For n = 0,

24.11

Y₀(x) = ₂/^π (ln (x/2) + γ)J₀(x) + ₂/^π[x² - _x⁴/^2⋅2 + x⁶/3⋅3 +24⋅64(1 + ₁/² + ₁/³) - ...]

24.12

Y_-n(x) = (-1)ⁿY_n(x)

n = 0, 1, 2, ...

For any value n >(not less than) 0, J_n(x) is bounded at x=0 while Y_n(x) is unbounded.

GENERAL SOLUTION OF BESSEL'S DIFFERENTIAL EQUATION

24.13

y = AJ_n(x) + BJ_-n(x)

x not = 0, 1, 2, ...

24.14

y = Aj_n(x) + By_n(x)

all n

24.15

y = AJ_n(x) + B_n(x)

∫ xⁿ J_-n(xc)

all n

where A and B are arbitrary constants.

GENERATING FUNCTION FOR J_n(x)

24.16

e^x(t-(1/t))/2 = ∑_n=-∞^∞ J_n(x)tⁿ

RECURRENCE FORMULAS FOR BESSEL FUNCTIONS

24.17

J_n+1(x) = _2n/^x J_n(x)

J_n-1(x)

24.18

J'_n(x) = ₁/²[J_n-1(x) - J_n+1(x)]

24.19

xJ'_n(x) = xJ_n-n(x) - nJ_n(x)

24.20

xJ'_n(x) = nJ_n(x) - xJ_n+1(x)

24.21

d/dx(xⁿJ_n(x)) = xⁿJ_n-1(x)

24.22

d/dx(x^-nJ_-n(x)) = -x_-nJ_n+1(x)

The functions Y_n(x) satisfy identical relations.

BESSEL FUNCTIONS

DIFFERENTIAL EQUATION FOR Ber, Bei, Ker, Kei FUNCTIONS

24.72

x²y′′ + xy′ - (ix² + n²)y = 0

The general solution of this equation is

24.73

y = A{Ber_n(x) + i Bei_n(x)} + B{Ker_n(x) + i Kei_n(x)}

GRAPHS OF BESSEL FUNCTIONS

Fig. 24-1

Fig. 24-2

Fig. 24-3

Fig. 24-4

Fig. 24-5

Fig. 24-6

25 LEGENDRE FUNCTIONS

LEGENDRE'S DIFFERENTIAL EQUATION

25.1

(1 - x²)y" - 2xy' + n(n + 1)y = 0

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

LEGENDRE POLYNOMIALS

If n = 0, 1, 2, ..., solutions of 25.1 are Legendre polynomials P_n(x) given by Rodrigues's formula

25.2

P_n(x) = ¹/_2ⁿn! ^dn/_dxⁿ(x² - 1)ⁿ

SPECIAL LEGENDRE POLYNOMIALS

25.3 P₀(x) = 1

25.4 P₁(x) = x

25.5 P₂(x) = ¹/₂(3x² - 1)

25.6 P₃(x) = ¹/₂(5x³ - 3x)

25.7 P₄(x) = ¹/₈(35x⁴ - 30x² + 3)

25.8 P₅(x) = ¹/₈(63x⁵ - 70x³ + 15x)

25.9 P₆(x) = ¹/₁₆(231x⁶ - 315x⁴ + 105x² - 5)

25.10 P₇(x) = ¹/₁₆(429x⁷ - 693x⁵ + 315x³ - 35x)

LEGENDRE POLYNOMIALS IN TERMS OF θ WHERE x = cos θ

25.11 P₀(cos θ) = 1

25.12 P₁(cos θ) = cos θ

25.13 P₂(cos θ) = ¹/₂(3 cos² θ - 1)

25.14 P₃(cos θ) = ³/₂(5 cos θ - 3 cos³ θ)

25.15 P₄(cos θ) = ¹/₈(9 + 20 cos² θ - 35 cos⁴ θ)

25.16 P₅(cos θ) = ¹/₁₂₈(30 cos θ + 8 - 35 cos³ θ + 63 cos⁵ θ)

25.17 P₆(cos θ) = ¹/₃₂(5θ + 105 cos² θ + 126 cos⁴ θ + 231 cos⁶ θ)

25.18 P₇(cos θ) = ¹/₅₁₂(1715 cos θ + 189 cos θ + 231 cos θ + 429 cos θ

GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS

25.19

¹/_{√(1 - 2tx + t²)} = ^∞∑_n=0 P_n(x)tⁿ

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