Tetration Forum

Hi,

Does anyone have an idea how to solve the following functional equation?

Thanks

$f(a^x) = f(x) \cdot a^x ln(a)$

$f(x)=f(x^1)=f(1)x\ln(x)$
Let's start there. Then, we plug in 1 to get $f(1)=0$ . Then $f(x)=0$ .

(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.

kobi_78

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