was asked to find out the value of $f(\sqrt{2})$

Question

I was given $f(1+{\sqrt{2}i\over n})=-{2\over n^2}$ where $f$ analytic from $|z|<3\to\mathbb{C}$. and was asked to find out the value of $f(\sqrt{2})$

I defined $g(z)=f(1+z)-z^2$ and then got $f(1+z)=z^2$ by Identity Theorem and then just put $f(z)=(z-1)^2$ and then $f(\sqrt{2})=3-2\sqrt{2}$

am I right in every step?

Answer 1

Yes, that's correct, and the point of the question seems to be exactly that use of the Identity Theorem. Since the set $1+\frac{\sqrt{2}i}{n}$ has the accumulation point $1$ in the domain $|z| < 3$, and $f$ agrees with $(z-1)^2$ on those points, then $f(z)=(z-1)^2$.