Please help with this problem:
let f be an entire function such that f(5z) = 5f(z). Prove that f is linear.
Thanks
If f is entire than it has a Taylor series as it's Laurent series. Then what can be said about the expression (just write out a few terms on each side and equate coefficients and then figure what they have to be to obtain equality):
$\displaystyle \sum_{n=0}^{\infty} a_n(5z)^n=5\sum_{n=0}^{\infty}a_n z^n$
Thank you for your response. I still have 3 questions for this problem. Please help. Thank you.
1. When I write the terms out, do I match term by term?
2. the coefficient a_n on the left side is the same as the coefficient on the right side?
3. Do I need to use the chain rule to find a_n for f(5z)?
Merely note that since both the LHS and the RHS converge we may rewrite shawsend's equation as $\displaystyle \sum_{n=1}^{\infty}z^n\left(5a_n-5^na_n\right)=0$. Clearly though for the LHS to equal zero we need that $\displaystyle 5a_n-5^na_n=0\quad \forall n$. It is trivially true for $\displaystyle n=1$ for $\displaystyle n>1$ it will only be true if $\displaystyle a_n=0$. Therefore $\displaystyle f(z)=5\sum_{n=1}^{\infty}a_nz^n=5a_1z$