I came across with the logarithm of a formal power series in a paper i am reading, but i have not found what is the definition of this 'operator(?)'. Can someone please let me know the definition and if the log obeys the same rules as in calculus?
I must add that I have this particular example:
Based on Mike answer, why is that in the second equality there is not a $(-1)^{k-1}$ term in the inner summation?
I'll really appreciate it. Thanks
Best Answer
In general, if $F(x)=\sum_{n\ge 0} f_nx^n$ is a formal power series, and if $G(x)=\sum_{\color{blue}{n\ge 1}}g_nx^n$ is a power series for which $G(0)=0$, then the composite power series $(F\circ G)(x)$ is defined as $$ (F\circ G)(x)= f_0+f_1\cdot G(x)+f_2 [G(x)]^2+f_3[G_3(x)]^3+\dots $$ The condition $G(0)=0$ is necessary for this to be a well-defined power series.
You can apply this with $F(x)=\log(1+x)=\sum_{n\ge 1} (-1)^{n-1}\frac{x^n}n$ to get a power series representation for $\log(1+G(x))$, which gives meaning to take the $\log$ of $1+G(x)$. This will still satisfy the properties of $\log$, like $\log[(1+G)(1+H)]=\log(1+G)+\log(1+H)$.