# Linked Questions

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### $\int e^{-x^2}dx$ [duplicate]

Possible Duplicate: Proving $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$ How does one integrate $\int e^{-x^2}\,dx$? I read somewhere to use polar coordinates. How is this done? What ...
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Prove that $\int_{-\infty}^{\infty}e^{-z^2}dz=\sqrt{\pi}$. Here is my attempted solution: Define $a:=\sqrt{\pi}e^{\frac{\pi i}{4}}$ and let $f(z) = \frac{e^{-z^2}}{1+e^{-2az}}$. Note that $a^2=\... 2answers 106 views ### Strange integrand Is it possible, and if yes, how, to evaluate an integral like$\int \sqrt{x} e^{x}dx$? I have hear of the Gaussian function which integrates to$\sqrt{\pi}$but what about this? Thank you. 2answers 228 views ### Which is the easier way to do integration by parts when there is an exponential term? I am trying to calculate the following integral, and I would like to know if there is a general rule where we set either$u(x)$equal to the exponential term or$v'(x)$equal to the exponential term. ... 2answers 1k views ### How to compute the integral$\int_{-\infty}^\infty e^{-x^2}\,dx$? [duplicate] How to compute the integral$\int_{-\infty}^\infty e^{-x^2}\,dx$using polar coordinates? 1answer 428 views ### Prove the Wallis formula form$\left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}$How would you prove the following Wallis formula form $$\left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}?$$ Thanks in advance! 1answer 1k views ### Evaluating definite integrals This question came up when I was reading through this question. Are there definite integrals which cannot be computed using any real analysis techniques but are amenable using only complex analysis ... 1answer 311 views ### Evaluating$\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$Q: Suppose$\alpha>0$and$|\beta|<\pi/2, show that \begin{align*} \textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 \sqrt{\pi/\alpha}\... 1answer 201 views ### Value of an integral related to Stirling's formula Consider the following improper integral : $$I = \int_1^\infty \left(\{t\}-\frac{1}{2}\right)\frac{dt}{t}.$$ Comparing with Stirling's formula, we can see thatI = \ln(\sqrt{2\pi}) - 1$. Is there ... 1answer 211 views ###$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$Hi I am trying to calculate the sum given by $$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} \vartheta_3\big(-\frac{\pi\beta}{2\alpha},e^{-\pi^2/(... 1answer 593 views ### Computing the integral of e^{-x^2} over the entire line [duplicate] Possible Duplicate: Proving \\int_{0}^{+\\infty} e^{-x^2} dx = \\frac{\\sqrt \\pi}{2} At lunch with a math friend years ago, he showed me an integral whose solution was, he said, so beautiful ... 1answer 49 views ### Computing the integral \int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}\mathrm{d}x I am pretty stuck when I tried to calculate the Wigner function for the coherent state. Below is part of the equation that I find very challenging.$$ \int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}\mathrm{d}x\... 0answers 224 views ### Integral$\int_0^{\infty} \frac{e^{-x^2}}{a+b\cos{x}}dx$Hello there I am trying to solve for$a > b$: $$I=\int_0^{\infty} \frac{e^{-x^2}}{a+b\cos{x}}dx$$ My thought was to expand into fourier series $$g(t)=\frac{1}{a+b\cos t}$$ Since g(t) has the ... 0answers 62 views ### Suitable contour for an integral ($\Gamma(1/2)$) Consider the following integral $$\int_0^\infty \frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi}$$ This can be evaluated using contour integration methods. A similar question was asked before (unfortunately I ... 0answers 193 views ### Residue calculus:$\int_{-\infty}^\infty e^{-x^2} \mathrm{d}x$[duplicate] I am pretty sure I have read the answer somewhere on this site, but sadly I am unable to find the question. How to evaluate$\int_{-\infty}^\infty e^{-x^2} \mathrm{d}x\$ using the residue theorem?

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