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I found this on Priestley's Complex Analysis in the Laplace transforms bit.

Suppose $f$ satisfies $f'(t)=f(kt)$ for $t>0$, where $0 and $f(0)=1$. Prove that $$f(t)=\sum_{n=0}^{\infty}\frac{k^{n(n-1)/2}}{n!}t^n$$

Applying the Laplace transform directly to $f'(t)=f(kt)$ gives a functional equation but I'm unsure how to solve it. Any hints?

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