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This is from an old qualifying examination question. Let $a>1$ be fixed. Show that $$ \displaystyle A_N=\pi i a \int_1^N t^{a-\frac{3}{2}}e^{\pi i t^a} dt $$ converges to some complex number as $N \rightarrow \infty$. I tried using integration by parts, but no luck. Is there something that I'm missing here?. Any hints and comments are greatly appreciated. |
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It looks like an integration by parts should do the job, though. Setting $u=t^{-1/2}$ and $dv=t^{a-1}e^{i\pi t^a}dt$, you get: $$ A_N= N^{-1/2}e^{i\pi N^a} +1 + \int_1^N \frac{1}{2} t^{-3/2}e^{i\pi t^a}dt. $$ Now it is clear that this converges to $$ 1+\int_1^{+\infty} \frac{1}{2} t^{-3/2}e^{i\pi t^a}dt. $$ In particular, note that the last integral converges absolutely since $3/2>1$. Edit: I have just seen your last comment. For the first term, we have $|N^{-1/2}e^{i\pi N^a}|=1/\sqrt{N}\rightarrow 0$ as $N\rightarrow+\infty$. |
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