I need to calculate the asymptotics of the integral $$\left(\int_0^1 \mathrm e^{-tx} f(t)\right)^j$$ for $x\to\infty$. I suspect (and would like to prove), that this behaves like $$\left(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{x^{n+1}}\right)^j,$$ but I'm not sure how. I get this result if I give each of the integrals a different parameter $x_i$, use Watson's lemma on each of them seperatly (which I can because the asymptotic integral is finite for finite $x_i$) and then evaluate at $x_1 = \ldots = x_j$. Does that make sense?
Can Watson's Lemma be applied on multiple integrals simultaneously?
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asymptotics