A functional equation
#1
Hi,

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

Thanks

\( f(a^x) = f(x) \cdot a^x ln(a) \)
#2
\( f(x)=f(x^1)=f(1)x\ln(x) \)
Let's start there. Then, we plug in 1 to get \( f(1)=0 \). Then \( f(x)=0 \).
#3
(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.


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