Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity.
(a) Find the power series expansion for f'
(b) Where does it converge?
(c) Find the power series expansion for f^2
(d) Where does it converge?
(e) Suppose that f'(x)^2 + f(x)^2 = 1, f(0) = 0, f'(0) = 1
Find a_0, a_1, a_2, a_3, a_4, a_5
The Attempt at a Solution
Hi everyone, here's my attempt. I'm not sure if it's right though, so I would really appreciate it if you could please take a look at it:
(a) f'(z) = ∑j.a_j.z^(j-1), sum from 0 to infinity
= ∑j.a_j.z^(j-1), sum from 1 to infinity
= ∑(j+1).a_(j+1).z^j, sum from 0 to infinity
(b) We don't what the values of a_j are, so we can only say with definity that it converges for z = 0
(c) f^2 = (a_0)^2 + (a_0.a_1)z + (a_0.a_2)z^2 + ...
+ (a_1.a_0)z + (a_1.a_1)z^2 + ... etc.
(d) Again, we can only say it converges for definite where z = 0.
(e) f(0) = 0
a_0 + a_1(0) + a_2(0) +... = 0
i.e. a_0 = 0
f'(0) = 1
a_1 + 2.a_2.z + 3.a_3.z^2 +.... = 1
The fact it is equal to 1 means it converges, but, as above, this can only happen if z = 0
i.e. a_1 = 1
Calculate f'(x)^2 = 1 + z(4.a_2) + z^2(6.a_3 + 4.a_2.a_2) + ...
and f(x)^2 = z^2(a_1.a_1) + z^3(2.a_1.a_2) + ...
The coefficients of the z terms in f'(x)^2 must be equal to minus the coefficients of the corresponding z terms in f(x)^2.
So we find that:
a_2 = 0
a_3 = -1/6
a_4 = 0
a_5 = 1/120
Regarding the convergence in (b) and (d), am I correct in what I say, in that there is no way of knowing what the a_j's are, so the only way we can know the series converge is when z = 0?
Also, have I multiplied the series correctly?
I'm sure I must have done something wrong, as it's a question from a past paper in my complex analysis class, but I don't seem to have used any complex analysis here...
Thanks for any help!