I have to solve this integral using Cauchy's integral formula. I tried to substitute it with several different attempts but without a solution. Can anyone help? $$\int_0^{2\pi} \frac{1}{a^2 \cdot {\cos}^2(t) + b^2 \cdot {\sin}^2(t)} dt$$
Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b&gt;0$.   My Try:  So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\pi\}$.   On $\Gamma$, $z=a\cos t+i b \sin t$, so $|z|^2=z\bar{z}=a^2\cos^2 t+b^2 \sin^2 t$.   Now, $dz=-a\sin t+i b \cos t dt$.   Hence, the integral becomes $\displaystyle \int_\Gamma \frac{dz}{z(iab+(\sin t \cos t)(b^2-a^2))}$. I know that $\displaystyle \int_\Gamma \frac{dz}{z}=2\pi i$. Now, how do I get rid of $\sin t \cos t$ part? I am stuck here. Can somebody please explain how?
Why is $\sqrt {12} = 2 \sqrt 3$? Evaluate $\prod \frac {2k - 1} {2k}$ I came across to this: evaluate $$\prod_{k = 1}^{n} \frac {2k - 1} {2k}.$$  I brought it to a closed-form expression in asymptotics but I suspect my asymptotics is bad and I also want to see different solutions, hence the question.  My original solution: Write   $$\begin {eqnarray} \prod_{k = 1}^{n} \frac {2k - 1} {2k} &amp; = &amp; \exp \left (\sum_{k = 1}^{n} \log \left (1 - \frac {1} {2k}\right) \right) \nonumber \\ &amp; = &amp; \exp \left (- \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m}\right). \end {eqnarray}$$  Let $$S = \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} \tag {1}.$$ Multiply both sides of $(1)$ by $2k$ to get $2k S &lt; S + 1$, and by $4k$ to get $4kS &gt; S + 2$. Hence, $$\frac {2} {4k - 1} &lt; S &lt; \frac {1} {2k - 1}.$$ Then, $$\frac {1} {2} \left( \log (4n - 1) - \log 3 \right) - 2 + O \left(\frac {1} {n^2}\right) &lt; \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} &lt; \frac {1} {2} \log (2n - 1) - 1 + O \left(\frac {1} {n^2}\right).$$ Hence, $$\sqrt {\frac {c} {2n - 1}} &lt; \prod_{k = 1}^{n} \frac {2k - 1} {2k} &lt; \sqrt {\frac {3 c^2} {4n - 1}},$$ where $c$ is a constant. Any better approach? Remove space before punctuation I need to remove locally spaces arround (especially before) punctuation signs. This spaces are added by french polyglossia.  \documentclass{article} \usepackage{fontspec} \usepackage{polyglossia} \setmainlanguage{french}  \newcommand{\nospace}[1]{\texttt{#1}}  \begin{document} \nospace{a:b} \end{document}     Thanks for help. 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Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places? Benefits of using QQ-plots over histograms In , Nick Cox wrote:     Binning into classes is an ancient method. While histograms can be useful, modern statistical software makes it easy as well as advisable to fit distributions to the raw data. Binning just throws away detail that is crucial in determining which distributions are plausible.   The context of this comment suggests using QQ-plots as an alternative means to evaluate the fit. The statement sounds very plausible, but I'd like to know about a reliable reference supporting this statement. Is there some paper which does a more thorough investigation of this fact, beyond a simple “well, this sounds obvious”? Any actual systematic comparisons of results or the likes?  I'd also like to see how far this benefit of QQ-plots over histograms can be stretched, to applications other than model fitting. Answers on  agree that “a QQ-plot […] just tells you that "something is wrong"”. I am thinking about using them as a tool to identify structure in observed data as compared to a null model and wonder whether there exist any established procedures to use QQ-plots (or their underlying data) to not only detect but also describe non-random structure in the observed data. References which include this direction would therefore be particularly useful. Can a sequence which decays more slowly still yield a converging series? 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