A functional equation - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: A functional equation (/showthread.php?tid=470) A functional equation - kobi_78 - 07/05/2010 Hi, Does anyone have an idea how to solve the following functional equation? Thanks $f(a^x) = f(x) \cdot a^x ln(a)$ RE: A functional equation - 73939 - 07/08/2010 $f(x)=f(x^1)=f(1)x\ln(x)$ Let's start there. Then, we plug in 1 to get $f(1)=0$. Then $f(x)=0$. RE: A functional equation - bo198214 - 07/08/2010 (07/05/2010, 07:28 PM)kobi_78 Wrote: Hi, Does anyone have an idea how to solve the following functional equation? Thanks $f(a^x) = f(x) \cdot a^x ln(a)$ Thats called "Julia equation", generally f(F(x))=F'(x)f(x) to solve for f. You obtain an Abel function from it, or obtain it from an Abel function $\alpha$ via: $f(x)=\frac{1}{\alpha'}$ you can solve it at a fixed point via comparing the powerseries coefficients.