On the Padé approximant wiki, people present various form of padre approximant for $\sin{(x)}, \exp{(x)}$, erf$(x)$ but does not provide instruction on how to create them.
What would be the correct way to find the Padé approximant for $\ln{(1+x)}$ ?
Thank you very much !
I shall make the solution as simple as possible. It is simplistic but it works.
The $[n,m]$ Padé approximant write $$f(x)=\frac{\sum_{k=0}^n a_k\,x^k}{1+\sum_{k=1}^m b_k\,x^k}$$ But, you also have $$f(x)=\sum_{k=0}^\infty c_k\,x^k$$ Cross multiply $$\left(1+\sum_{k=1}^m b_k\,x^k\right)\left(\sum_{k=0}^\infty c_k\,x^k \right)=\sum_{k=0}^n a_k\,x^k$$
Expand and identify the coefficients $a_k$ and $b_k$. That is all !
