I asked this question: Holomorphic function such that $\theta\left(z+\omega_j\right)=a_j\theta\left(z\right)$ satisfies $\theta\left(z\right)=ae^{bz}$.
That question got closed because it was a duplicate of: Entire functions such that $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $
But I cannot finish the solution using the hints given in that second question. Basically, we want to show the following:
Given $\omega_1,\omega_2\in\mathbb{C}$ forming an $\mathbb{R}$-basis for $\mathbb{C}$, and given $a_1,a_2\in\mathbb{C}\setminus\left\{0\right\}$, prove that there exists $B\in\mathbb{C}$ such that $e^{B\omega_1}=a_1$ and $e^{B\omega_2}=a_2$.
To prove this, write $B=x+iy$, $\omega_1=a+bi$, $\omega_2=c+di$. The equation $e^{B\omega_1}=a_1$ becomes $e^{\left(ax-by\right)+i\left(bx+ay\right)}=a_1$, which gives $ax-by=\ln\left(\left |a_1\right |\right)$.
Similarly, $cx-dy=\ln\left(\left |a_2\right |\right)$.
These two equations are linearly independent (because $\omega_1,\omega_2$ form an $\mathbb{R}$-basis for $\mathbb{C}$), so they already determine $x$ and $y$, and I have no control on the argument of $e^{B\omega_j}$. So I do not understand how to find my $B$.
Maybe using that there is an entire function $\theta$ such that $\theta\left(z+\omega_j\right)=a_j\theta\left(z\right)$ for all $j\in\left\{1,2\right\}$ allows us to find a relation between $a_1$ and $a_2$ which helps us to find our desired $B$?