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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.