Thread: Prove an entire function is linear

1. Prove an emtire function is linear

let f be an entire function such that f(5z) = 5f(z). Prove that f is linear.

Thanks

2. 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):

$\sum_{n=0}^{\infty} a_n(5z)^n=5\sum_{n=0}^{\infty}a_n z^n$

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)?

4. Originally Posted by conmeo

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)?
Yes, yes, no. Just write:

$a_0+5 a_1 z+25 a_2 z^2+\cdots=5(a_0+a_1 z+a_2 z^2+\cdots)$ and in order for that to hold, all the $a_n$ have to be zero except $a_1$. That implies the function is linear.

5. since it's a Taylor series, Do I have to write the n! in the denominator?

6. Originally Posted by conmeo
since it's a Taylor series, Do I have to write the n! in the denominator?
Merely note that since both the LHS and the RHS converge we may rewrite shawsend's equation as $\sum_{n=1}^{\infty}z^n\left(5a_n-5^na_n\right)=0$. Clearly though for the LHS to equal zero we need that $5a_n-5^na_n=0\quad \forall n$. It is trivially true for $n=1$ for $n>1$ it will only be true if $a_n=0$. Therefore $f(z)=5\sum_{n=1}^{\infty}a_nz^n=5a_1z$