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I'm trying to prove the following:

Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$.

I've tried to bound it in a straight forward way to no avail. Also, assuming the convergence is among real positive epsilons, I could reduce it to the purely real case ($\Im z$ does not matter) and apply Dini's theorem. However, it does not seem to work well for arbitary $\varepsilon$.

How to prove it for the general complex case?

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