Methods 4

Question 1

Regular Singular Point

Answer 1

For a general linear second-order differential equation for w(z), of the form w”(z) + p(z)w’(z) + q(z)w(z) = 0

A point z=z0 is a regular singular point of it is not an ordinary point, but both (z-z₀)p(z) and (z-z₀)²q(z) are analytic at z₀

Question 2

Ordinary Point

Answer 2

For a general linear second-order differential equation for w(z), of the form w”(z) + p(z)w’(z) + q(z)w(z) = 0

A point z=z₀ is an OP if both p(z) and q(z) are analytic at z₀

Question 3

Periodic Sturm Liouville Systems

Answer 3

Periodic Boundary Conditions

y(a) = y(b) 
y'(a) = y'(b) 

Where y(x) is a periodic function defined on a

Question 4

Regular Sturm Liouville

Answer 4

P(x) and Q(x) singular at x = +- 1

Question 5

D’Alembert’s Ratio Test

Answer 5

Radius of convergence - property for Legendre polynomials

If regular singular points at x=+-1 - the radius of convergence if likely to be UNITY

Confirm using D’Alembert’s Ratio test
-Diverge for |z|>1 : D’Alembert’s ratio test indicates divergence

Lim|(bk+1Z^2k+2)/bkZ^2k)>1 or lim|bk/bk+1| as k tends to infinity

Question 6

Eigwnvalue are real

Answer 6

Define the eigenvalue and eigenfunction solutions to the problem as:
Lyk = -λk yk

1. = = -λk
2. = = = complex conjugate of

Comparing 1 and 2: clear λk = conjugate λk

So λk are real

Question 7

Eigenfunctions Yj(x) Yk(x) corresponding to eigenvalues λj, λk are orthogonal with respect to inner product

Answer 7

Let λj, λk be distinct eigenvalues, L is self-adjoint, λj ≠ λk

1. w = w = -λj w
2. w = w = w = -λk as λk are real

Subtracting 2 from 1:

Question 8

Adjoint

Answer 8

The operator A is said to be self joint if: A’ = A

Question 9

Legendre Equation

Answer 9

(1-x²)y” -xy’ +λy = 0 where λ = v(v+1)

• index up
• use naive
• Eigenvalues: λk =k²
• Eigenfunctions: Tk(x)

Weight function: use integrating factor
1/√(1-x²)

Question 10

Bessel Equation

Answer 10

x²w” +xw’ +(x²-v²)w = 0

W(z) =
AJv(z) + BJ-v(x) v>0
AJ0(z) +BY0(z) v=0
AJm(z) + BYm(z) v=m

Indicial: c²-v²

Recurrence: odd terms = 0
Make a2k into bk