Integration of cotangent with complex calculus
I am supposed to integrate the function $f(t^{\prime})=cot(2 w t^{\prime})$ by taking into account the singularities at $t^{\prime}=\frac{n\pi}{2w}$ via complex integration. It is quite easy to calculate the integral via substitution, i.e., $$ \int_{0}^tdt^{\prime}\, cot(2 w t^{\prime})=\int_{u(0)}^{u(t)}du\, \frac{1}{2wu}=\frac{1}{2w}ln(sin(2wt))\, . $$ Now, with complex calculus, as I understood (I haven't used complex calculus yet), one can integrate by constructing a contour that circumvents the singularities. The contour, I would think of would be from $0$ to $t$ and then around a quarter of a circle, from $t$ to $it$ and then back from $it$ to $0$. However, there are in between arbitrary number of poles, depending on how large $t$ is. How can I deal with this and how can I parametrise, when integrating around the poles?
$\endgroup$ 21 Answer
$\begingroup$As mentioned by previous comments, complex methods generally work solely for definite integrals. Specifically, to evaluate real integrals through contours you must set the bounds, for instance $\infty$ to $-\infty$.
Keep in mind, when evaluating real integrals through contour integration we use some manipulation of $$ \oint_C = \int_{-R}^{R} + \oint_{C_R}$$ to solve for the $ \int_{-R}^{R}$ component through evaluating the contour integrals $\oint_C$ and $\oint_{C_R}$, where $C$ is the contour, $C_R$ is the semicircle portion of $C$, and the real integral represents the portion of $C$ with no imaginary part. You mention using a quarter-circle contour, but it is often easier to use a semicircle contour and simplify to your specified bounds using properties of odd/even functions.
$\oint_C $ can be evaluated through various methods, such as Cauchy's Integral Formula or using the residues of the singularities. $\oint_{C_R}$ can be shown to approach $0$ when $R \rightarrow \infty$, through either using the ML approximation method or using a polar substitution $z = Re^{i\theta}$ from $0$ to $\pi$. When you set $R$ to an arbitrary $t$, $\oint_{C_R}$ cannot be shown to approach $0$, and therefore we cannot proceed for solving $\int^{R}_{-R}$.
Thus, if you wish to evaluate something such as $$ \int_{0}^{\infty}dt' cot(2wt') $$ through methods of contour integration, we can use $$cot(z) = i\frac{e^z+e^{-z}}{e^z-e^{-z}} $$ by Euler's identity. However, you cannot solve the original integral provided without definitive bounds.
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