Linked Questions

1
vote
3answers
2k views

$\int e^{-x^2}dx$ [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 ...
6
votes
2answers
4k views

Complex Analytic Proof of the Gaussian Integral $\int_{-\infty}^{\infty}e^{-z^2}dz=\sqrt{\pi}$

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=\...
2
votes
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.
1
vote
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. ...
0
votes
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?
11
votes
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!
9
votes
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 ...
5
votes
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}\...
5
votes
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 that $I = \ln(\sqrt{2\pi}) - 1$. Is there ...
3
votes
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/(...
1
vote
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 ...
0
votes
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\...
5
votes
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 ...
1
vote
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 ...
1
vote
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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