16.6 EXERCISES
(c) Determine the radius of convergenceRof theσ= 3 series and relate it to
the positions of the singularities of Legendre’s equation.16.6 Verify thatz= 0 is a regular singular point of the equation
z^2 y′′−^32 zy′+(1+z)y=0,
and that the indicial equation has roots 2 and 1/2. Show that the general solution
is given byy(z)=6a 0 z^2∑∞
n=0(−1)n(n+1)2^2 nzn
(2n+3)!+b 0(
z^1 /^2 +2z^3 /^2 −z^1 /^2
4∑∞
n=2(−1)n 22 nzn
n(n−1)(2n−3)!)
.
16.7 Use the derivative method to obtain, as a second solution of Bessel’s equation
for the case whenν= 0, the following expression:
J 0 (z)lnz−∑∞
n=1(−1)n
(n!)^2(n
∑r=11
r)
(z2) 2 n
,given that the first solution isJ 0 (z), as specified by (18.79).
16.8 Consider a series solution of the equation
zy′′− 2 y′+yz=0 (∗)
about its regular singular point.
(a) Show that its indicial equation has roots that differ by an integer but that
the two roots nevertheless generatelinearly independent solutionsy 1 (z)=3a 0∑∞
n=1(−1)n+1 2 nz^2 n+1
(2n+1)!,
y 2 (z)=a 0∑∞
n=0(−1)n+1(2n−1)z^2 n
(2n)!.
(b) Show thaty 1 (z)isequalto3a 0 (sinz−zcosz) by expanding the sinusoidal
functions. Then, using the Wronskian method, find an expression fory 2 (z)in
terms of sinusoids. You will need to writez^2 as (z/sinz)(zsinz) and integrate
by parts to evaluate the integral involved.
(c) Confirm that the two solutions are linearly independent by showing that
their Wronskian is equal to−z^2 , as would be expected from the form of (∗).16.9 Find series solutions of the equationy′′− 2 zy′− 2 y= 0. Identify one of the series
asy 1 (z)=expz^2 and verify this by direct substitution. By settingy 2 (z)=u(z)y 1 (z)
and solving the resulting equation foru(z), find an explicit form fory 2 (z)and
deduce that
∫x
0e−v2
dv=e−x2 ∑∞
n=0n!
2(2n+1)!(2x)^2 n+1.16.10 Solve the equation
z(1−z)d^2 y
dz^2+(1−z)dy
dz+λy=0as follows.
(a) Identify and classify its singular points and determine their indices.