I am studying a paper by Rosay and Rudin, titled Holomorphic maps from $\mathbb{C}^n$ to $\mathbb{C}^n$, and at some point they assume the existence of an entire function $f$ in $\mathbb{C}$ satisfying the following properties:
$e^{f(0)} = \frac{1}{\alpha}$, $f'(0) = 0$, $f(1) = 0$, $f'(1) = \frac{1+\alpha^2}{1-\alpha^2}$,
where $\alpha\in\mathbb{C}$, $0<|\alpha|<1$. I do not think that they are wrong but I can not find such an entire function. Any ideas?
It seems to me like they only specify $f(0),f'(0),f(1),f'(1)$. Getting those 4 conditions down can be done in multiple ways but perhaps most obvious way is just taking a polynomial of order 3.
If you don't know how to do that look at Lagrange interpolation.