# Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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### Trouble with a Fourier Transform

I'm getting some conflicting results when trying to take this integral : $$\int^{L/2}_{-L/2}xe^{i\frac{2\pi}{L}(n-n')x}dx$$ where $n$ and $n'$ are both integers. I'm having ...
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### Which zeros of $z^4+6z^2+13=0$ are in the upper half plane?

I want to calculate $\int_0^{+\infty}\frac{x^2}{x^4+6x^2+13} dx$ using residue theorem. Consider $f(z)=\frac{z^2}{z^4+6z^2+13}$, so I need to find the zeros of $z^4+6z^2+13=0$ which are in the upper ...
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• 461
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• 2,251
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### Trying to solve $I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx$

While trying to find an answer to this problem on the forum, I came across this integral: $$I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$ Where $c$ and $k$ are real numbers. I ...
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### complex analysis - Help with integrating $\int_0^{\infty} \frac{(\log x)^4}{x^2 + 1} \operatorname d\!x$ [duplicate]

I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane): I have split the contour into 4 parts - the large semi-circle, ...
1 vote
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### a limit of a complex function of $\zeta(s)\zeta(s+1)\Gamma(s)$

$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s)$$ This has a double pole at $s=0$ , from $\zeta(s+1)$ and $\Gamma(s)$ respectively, and $\lim_{s\to0}s^2f(s) = -1/2$ Then, the next step, I have difficulty with ...
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### Proving a theorem related to uniqueness of Fourier Series for complex-valued function.

I was studying Fourier Analysis: An Introduction by Stein and Shakarchi. I came across this theorem in Section 2.2, page number 39 Theorem 2.1 Suppose that $f$ is an integrable function on the circle ...
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### Lebesgue Integral on Rough Paths

In a nutshell: how can we integrate rough paths using Lebesgue integration? What I mean by a rough path is a continuous map $[a,b]\to\mathbb{R}^n$, or $[a,b]\to\mathbb{C}$ in the complex case. (The ...
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