I am trying exercises in elliptic functions from Tom M Apostol and I am unable to think about this problem.
Problem is --> Prove that every elliptic function for can be expressed in the form for(z) = $R_1[\wp(z) ] $ + $ \wp'(z) R_2[\wp(z) ] $ , where $R_1$ and $ R_2$ are rational functions and $\wp $ has same set of periods as f.
I thought that as $\wp(z) $ is even so it's derivative would be odd function and every function can be written as combination of odd and even functions. Also, every even elliptic function is a rational function of $\wp(z) $ is a known result.
But I am not clear how to use all these results to prove this and whether these results are sufficient to prove this. Can someone please help?
It's just separating a function into its odd an even parts. If $f$ is elliptic then $$f(z)=\frac{f(z)+f(-z)}{2}+\frac{f(z)-f(-z)}{2}$$ is the sum of an even and odd elliptic function. The even part is a rational function of $\wp(z)$. Dividing the odd part by $\wp'(z)$ gives an even elliptic function which is also a rational function of $\wp(z)$. So the odd part has the form $\wp'(z)R_2(\wp(z))$ where $R_2$ is a rational function.