A complex integral

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A complex integral

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Gauler

#1 h Mar 30, 2025, 3:26 PM

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For $s=c+it$ with $|t|\le T$ , prove that
$\frac{1}{2\pi i}\int _{c-iT}^{c+iT}\frac{e^{su}}{s}ds=\frac{1}{2}+\frac{Tu}{\pi }+O\left(T^2u^2+\frac{c}{T}\right)$ for $T|u|\le 1$ .

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#2 h Mar 30, 2025, 4:30 PM

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Let s = c + it, so ds = i dt.
Then the integral becomes:

(1 / 2πi) ∫ from c - iT to c + iT of (e^(su) / s) ds
= (1 / 2π) ∫ from –T to T of [e^(cu) * e^(i t u) / (c + i t)] dt

Now use a known result:
(1 / 2π) ∫ from –T to T of [e^(i t u) / (c + i t)] dt
≈ 1/2 + (Tu) / π + error
where the error is about T² u² + c / T
(valid when T|u| ≤ 1)

So multiply everything by e^(cu)
(but e^(cu) ≈ 1 when u is small)

Final answer:
(1 / 2πi) ∫ = 1/2 + (Tu) / π + O(T² u² + c / T)

By the way.. can anyone tell me why people ask these questions? Is it for school, work, etc? Or for fun.

This post has been edited 1 time. Last edited by paxtonw, Mar 30, 2025, 4:32 PM

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