[PPT] - Bessel Functions Bernd Schr oder logo1 Bernd Schr oder PowerPoint Presentation

SLIDE 1

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions

Bernd Schr¨

der

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 2

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Why are Bessel Functions Important?

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 3

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Why are Bessel Functions Important?

1. Parametric Bessel equations

x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 arise when the equations ∆u = k∂u ∂t and ∆u = k∂ 2u ∂t2 are solved with separation of variables in polar or cylindrical

coordinates. The function y(r) describes the radial part of

the solution.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 4

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Why are Bessel Functions Important?

1. Parametric Bessel equations

x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 arise when the equations ∆u = k∂u ∂t and ∆u = k∂ 2u ∂t2 are solved with separation of variables in polar or cylindrical

coordinates. The function y(r) describes the radial part of

the solution.

2. Because 0 is a regular singular point of the equation, it is

natural to attempt a solution using the method of Frobenius.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 5

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 6

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 7

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 8

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 9

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 10

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 11

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 12

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 13

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 14

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 15

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 16

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 17

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 18

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 19

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 20

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 21

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r = 0

r(r −1)c0 +rc0 −ν2c0
xr

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 22

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r = 0

r(r −1)c0 +rc0 −ν2c0
xr +
(r +1)rc1 +(r +1)c1 −ν2c1
xr+1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 23

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r = 0

r(r −1)c0 +rc0 −ν2c0
xr +
(r +1)rc1 +(r +1)c1 −ν2c1
xr+1

+

∞

∑

k=2

(k +r)(k +r −1)ck +(k +r)ck +λ 2ck−2 −ν2ck
xk+r

= 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 24

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Frobenius Solution for x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 x2

∞

∑

n=0

cn(n+r)(n+r−1)xn+r−2+x

∞

∑

n=0

cn(n+r)xn+r−1+

λ 2x2−ν2 ∞

∑

n=0

cnxn+r = 0

∞

∑

n=0

(n+r)(n+r−1)cnxn+r+

∞

∑

n=0

(n+r)cnxn+r+

∞

∑

n=0

λ 2cnxn+r+2−

∞

∑

n=0

ν2cnxn+r = 0

∞

∑

k=0

(k+r)(k+r−1)ckxk+r+

∞

∑

k=0

(k+r)ckxk+r+

∞

∑

k=2

λ 2ck−2xk+r−

∞

∑

k=0

ν2ckxk+r = 0

r(r −1)c0 +rc0 −ν2c0
xr +
(r +1)rc1 +(r +1)c1 −ν2c1
xr+1

+

∞

∑

k=2

(k +r)(k +r −1)ck +(k +r)ck +λ 2ck−2 −ν2ck
xk+r

= 0

r2−ν2

c0xr+

(r+1)2−ν2

c1xr+1+

∞

∑

k=2

(k+r)2−ν2

ck+λ 2ck−2

xk+r = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 25

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 26

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 27

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 28

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν For r = ν we obtain c1

(ν +1)2 −ν2

= 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 29

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν For r = ν we obtain c1

(ν +1)2 −ν2

= 0, c1 = 0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 30

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν For r = ν we obtain c1

(ν +1)2 −ν2

= 0, c1 = 0 Recurrence relation for k ≥ 2:

(k +ν)2 −ν2

ck +λ 2ck−2 =

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 31

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν For r = ν we obtain c1

(ν +1)2 −ν2

= 0, c1 = 0 Recurrence relation for k ≥ 2:

(k +ν)2 −ν2

ck +λ 2ck−2 = ck = − λ 2 (k +ν)2 −ν2ck−2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 32

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Indicial Roots, Recurrence Relation

r2 −ν2 = 0, r1,2 = ±ν For r = ν we obtain c1

(ν +1)2 −ν2

= 0, c1 = 0 Recurrence relation for k ≥ 2:

(k +ν)2 −ν2

ck +λ 2ck−2 = ck = − λ 2 (k +ν)2 −ν2ck−2 = − λ 2 k(k +2ν)ck−2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 33

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 34

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 35

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2 = − λ 2 4n(n+ν)c2(n−1)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 36

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2 = − λ 2 4n(n+ν)c2(n−1) = (−1)2

λ 22

42n(n−1)(n+ν)(n−1+ν)c2(n−2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 37

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2 = − λ 2 4n(n+ν)c2(n−1) = (−1)2

λ 22

42n(n−1)(n+ν)(n−1+ν)c2(n−2) = (−1)3

λ 23

43n(n−1)(n−2)(n+ν)(n−1+ν)(n−2+ν)c2(n−3)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 38

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2 = − λ 2 4n(n+ν)c2(n−1) = (−1)2

λ 22

42n(n−1)(n+ν)(n−1+ν)c2(n−2) = (−1)3

λ 23

43n(n−1)(n−2)(n+ν)(n−1+ν)(n−2+ν)c2(n−3) . . . = (−1)n

λ 2n

4nn(n−1)···2·1·(n+ν)(n−1+ν)···(2+ν)(1+ν)c0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 39

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Even Numbered Terms

c2n = − λ 2 2n(2n+2ν)c2n−2 = − λ 2 4n(n+ν)c2(n−1) = (−1)2

λ 22

42n(n−1)(n+ν)(n−1+ν)c2(n−2) = (−1)3

λ 23

43n(n−1)(n−2)(n+ν)(n−1+ν)(n−2+ν)c2(n−3) . . . = (−1)n

λ 2n

4nn(n−1)···2·1·(n+ν)(n−1+ν)···(2+ν)(1+ν)c0 = (−1)n Γ(ν +1)

λ 2n

4nn!Γ(n+ν +1)c0

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 40

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 41

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

y1(x) =

∞

∑

n=0

c2nx2n+ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 42

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

y1(x) =

∞

∑

n=0

c2nx2n+ν =

∞

∑

n=0

(−1)n Γ(ν +1)

λ 2n

4nn!Γ(n+ν +1)x2n+ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 43

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

y1(x) =

∞

∑

n=0

c2nx2n+ν =

∞

∑

n=0

(−1)n Γ(ν +1)

λ 2n

4nn!Γ(n+ν +1)x2n+ν λ ν 2νΓ(ν +1)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 44

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

y1(x) =

∞

∑

n=0

c2nx2n+ν =

∞

∑

n=0

(−1)n Γ(ν +1)

λ 2n

4nn!Γ(n+ν +1)x2n+ν λ ν 2νΓ(ν +1) =

∞

∑

n=0

(−1)n λ 2n+ν 22n+νn!Γ(n+ν +1)x2n+ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 45

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

y1(x) =

∞

∑

n=0

c2nx2n+ν =

∞

∑

n=0

(−1)n Γ(ν +1)

λ 2n

4nn!Γ(n+ν +1)x2n+ν λ ν 2νΓ(ν +1) =

∞

∑

n=0

(−1)n λ 2n+ν 22n+νn!Γ(n+ν +1)x2n+ν =

∞

∑

n=0

(−1)n 1 n!Γ(n+ν +1)

λ x

2 2n+ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 46

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 47

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Jν(x) =

∞

∑

n=0

(−1)n 1 n!Γ(n+ν +1) x 2 2n+ν

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 48

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Jν(x) =

∞

∑

n=0

(−1)n 1 n!Γ(n+ν +1) x 2 2n+ν J−ν(x) =

∞

∑

n=0

(−1)n 1 n!Γ(n−ν +1) x 2 2n−ν .

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 49

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Jν(x) =

∞

∑

n=0

(−1)n 1 n!Γ(n+ν +1) x 2 2n+ν J−ν(x) =

∞

∑

n=0

(−1)n 1 n!Γ(n−ν +1) x 2 2n−ν . Both series are guaranteed to converge at least on (0,∞).

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 50

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 51

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

For λ > 0 and ν > 0 such that ν is not an integer, the general solution of the parametric Bessel equation x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 is y(x) = c1Jν(λx)+c2J−ν(λx).

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 52

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the First Kind

For λ > 0 and ν > 0 such that ν is not an integer, the general solution of the parametric Bessel equation x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 is y(x) = c1Jν(λx)+c2J−ν(λx). Of course, we are most interested in the solutions when ν is an integer.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 53

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the Second Kind

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 54

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the Second Kind

For ν not an integer we define Yν(x) := cos(νπ)Jν(x)−J−ν(x) sin(νπ) . For ν = m an integer we define Ym(x) := lim

ν→mYν(x).

The functions Yν are called the Bessel functions of the second kind.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 55

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Bessel Functions of the Second Kind

For ν not an integer we define Yν(x) := cos(νπ)Jν(x)−J−ν(x) sin(νπ) . For ν = m an integer we define Ym(x) := lim

ν→mYν(x).

The functions Yν are called the Bessel functions of the second

kind. For λ > 0 and any ν > 0 the general solution of the

parametric Bessel equation x2y′′ +xy′ +

λ 2x2 −ν2

y = 0 is y(x) = c1Jν(λx)+c2Yν(λx).

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 56

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 57

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

1. The shape of a vibrating drum membrane can be modeled

with the equation ∂ 2u ∂x2 + ∂ 2u ∂y2 = k∂ 2u ∂t2 and the condition that u is zero on the boundary.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 58

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

1. The shape of a vibrating drum membrane can be modeled

with the equation ∂ 2u ∂x2 + ∂ 2u ∂y2 = k∂ 2u ∂t2 and the condition that u is zero on the boundary.

2. u is the displacement from equilibrium of the particle at

(x,y) at time t.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 59

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

1. The shape of a vibrating drum membrane can be modeled

with the equation ∂ 2u ∂x2 + ∂ 2u ∂y2 = k∂ 2u ∂t2 and the condition that u is zero on the boundary.

2. u is the displacement from equilibrium of the particle at

(x,y) at time t.

3. The derivation is similar to that of the equation of an
scillating string. (Challenging exercise.)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 60

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

1. The shape of a vibrating drum membrane can be modeled

with the equation ∂ 2u ∂x2 + ∂ 2u ∂y2 = k∂ 2u ∂t2 and the condition that u is zero on the boundary.

2. u is the displacement from equilibrium of the particle at

(x,y) at time t.

3. The derivation is similar to that of the equation of an
scillating string. (Challenging exercise.)
4. For a round membrane, we solve the equation with

separation of variables in polar coordinates, which leads to the Bessel equation and Bessel functions.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 61

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 62

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions

SLIDE 63

logo1 Overview Solving the Bessel Equation Bessel Functions Application

Vibrating Drum Membranes (Outline)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bessel Functions