I have to maximize entropy and therefore formulated the dual function, calculated its derivatives, set them equal to zero and now I have to solve the following system analytically:
$$ e^{-\mu_1 - \mu_2} + e^{-\mu_1 - 2\mu_2} + e^{-\mu_1 - 3\mu_2} + e^{-\mu_1 - 4\mu_2} = e^{1} \\ e^{-\mu_1 - \mu_2} + 2e^{-\mu_1 - 2\mu_2} + 3e^{-\mu_1 - 3\mu_2} + 4e^{-\mu_1 - 4\mu_2} = e^{1} $$
I tried all kind of stuff but that didnt work out so far.
How can I solve this system analytically?
Mathematica gives this solutions:
$c_1,c_2\in \mathbb{Z}$
$ \left\{\mu_1=2 i \pi c_1+\log \left(-\frac{4 i \left(4 \sqrt{2}-7 i\right)}{81 e}\right),\mu_2=2 i \pi c_2+\log \left(i \left(\sqrt{2}+i\right)\right)\right\}$
OR
$ \left\{\mu_1= 2 i \pi c_1+\log \left(\frac{4 i \left(4 \sqrt{2}+7 i\right)}{81 e}\right),\mu_2=2 i \pi c_2+\log \left(-i \left(\sqrt{2}-i\right)\right)\right\}$