Questions tagged [gaussian-integral]
For questions regarding the theory and evaluation of the Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function $~e^{−x^2}~$ over the entire real line. . It is named after the German mathematician Carl Friedrich Gauss.
88
questions
217
votes
21
answers
157k
views
Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 8,008
4
votes
1
answer
1k
views
Generalized Owen's T function
As Wikipedia teaches us https://en.wikipedia.org/wiki/Owen%27s_T_function the Owen's T function $T(h,a)$ defines a probability of a bivariate event $X>h$ and $0<Y<a X$ where $X,Y$ are ...
- 11.4k
7
votes
1
answer
3k
views
Gaussian Integral using contour integration with a parallelogram contour
I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function,
$f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi z)}...
6
votes
1
answer
483
views
Polar Coordinate Transformation - Motivation
I am trying to work out the reason why the integral
$$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{e^{-(x^2+y^2)}}\,dx\,dy $$
is, in polar coordinates,
$$ \int_{-\infty}^{\infty}{e^{-r^2}} \,r\,...
- 1,059
30
votes
1
answer
2k
views
Why do zeta regularization and path integrals agree on functional determinants?
When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions.
The first ...
- 152k
48
votes
3
answers
21k
views
reference for multidimensional gaussian integral
I was reading on Wikipedia in this article about the n-dimensional and functional generalization of the Gaussian integral. In particular, I would like to understand how the following equations are ...
- 8,750
20
votes
2
answers
7k
views
Which functions are tempered distributions?
Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function
$$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$
where $\...
- 32.4k
16
votes
1
answer
17k
views
Entropy of the multivariate Gaussian
Show that the entropy of the multivariate Gaussian $N(x|\mu,\Sigma)$ is given by
\begin{align}
H[x] = \frac12\ln|\Sigma| + \frac{D}{2}(1 + \ln(2\pi))
\end{align}
where $D$ is the dimensionality of ...
- 1,133
12
votes
1
answer
18k
views
Show that integral of Gaussian distribution is 1
Under a normal distribution, μ = 0 and σ = 1, but when then integrating this equation, I get an error function.
Without using Riemann sums, how can I prove that this equation = 1?
I have only had a ...
- 123
10
votes
1
answer
1k
views
Integrating:$\int\limits_0 ^ {\infty}e^{-x^2}\ln(x)dx $
$$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx = -\frac{1}{4}(\gamma +2\ln(2))\sqrt{\pi} $$
This is a well known integral.
But I want to know how to solve it??
Also, please refrain using contour ...
- 447
10
votes
3
answers
1k
views
gaussian integral of power of cdf : $\int_{-\infty}^{+\infty} \Phi(x)^n \cdot \phi(a+bx) \cdot dx$
Is there an analytic solution for the following Gaussian integral?
$$\int_{-\infty}^{+\infty} \Phi(x)^n \cdot \phi(a+bx) \cdot dx$$
with
$n$, a positive integer (typically under 10)
$a,b$, real ...
- 101
3
votes
1
answer
176
views
Is there a double integral lurking in this proof?
There are many proofs that $\int_\mathbb{R}e^{-x^2}dx=\sqrt{\pi}$. What they all seem to have in common is a dependence on computing a double integral, at least in the sense of a product of two single ...
- 116k
2
votes
5
answers
272
views
Messy Gaussian Integral
I am trying to understand how to better perform the following integral.
$$\int^{\infty}_{0} x^4 e^{\frac{-x^2}{\beta^2}}\mathrm{d}x$$
I've done a little research and found that $e^{-x^2}$ doesn't ...
- 647
1
vote
2
answers
636
views
Proof of certain Gaussian integral form
I am having trouble understanding where the following integral form comes from:
$$\int_{-\infty}^{\infty} e^{-a x^2 }e^{-bx}=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}}$$ I see and understand that the value ...
- 13
1
vote
1
answer
481
views
Complex Gaussian Integral by change of variable
In this video (minute 5:20), they prove that
$$ I = \int_{-\infty}^{\infty}e^{i x^2}dx = e^{i\frac{\pi}{4}}\sqrt{\pi}$$
by deforming the integration contour in the complex plane from the real axis to ...
- 143
11
votes
3
answers
4k
views
Compute multivariate complex Gaussian integral
I don't know how to work out the homework of Leib&Loss P121, Ex4(b), in which we need to compute the following
$$
\int_{\mathbb{R}^n}\exp(-x^tAx)dx=\pi^{n/2}/\sqrt{\det A}
$$
where $A=A^t$ is a ...
- 1,461
10
votes
0
answers
370
views
Multivariable Integral, How to compute it?
Q
How to evaluate a multivariate integral with a Gaussian weight function?
$$
\mathcal{Z_{n}}
\equiv\int_{-\infty}^{\infty}
\exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\,
{\rm f}\left(x_{1},x_{2},\...
- 479
10
votes
1
answer
370
views
Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion
To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows:
I consider an experiment which consists of a continuum of trials ...
- 704
7
votes
3
answers
228
views
Log-convexity of a function defined by an integral (Normal Mills ratio)
Let's define $f(x)$, for all $x>0$ by :
$$f(x)=e^{x^2/2}\int_x^{+\infty}e^{-t^2/2}dt$$
I would like to prove that $f$ is log-convex, which is equivalent to the following condition :
$$\forall x>...
- 7,535
6
votes
2
answers
1k
views
Fourier transform of squared Gaussian Hermite polynomial
I'm attempting to calculate the first-order perturbation energy shift for the quantum harmonic oscillator with a perturbing potential of $V(x)=A\cos(kx)$. Omitting the relevant physical factors, I've ...
6
votes
3
answers
11k
views
Computing the integral $\int \exp(ix^2) dx$
I'm trying to compute the following integral: $I_1 = \int_{-\infty}^\infty \exp(iu^2) du$.
This is what I did, but it is wrong and I don't know why:
$$I_1^2 = \left (\int_{-\infty}^\infty \exp(iu^...
- 85
5
votes
2
answers
335
views
Prove that $9\int_{0}^{\infty}x^5e^{-x^3}\ln(1+x)dx=\Gamma\left({1\over 3}\right)-\Gamma\left({2\over 3}\right)+\Gamma\left({3\over 3}\right)?$
My last observance of this question by @Brightsun
Not so interesting integral but does have a neat closed form
$$9\int_{0}^{\infty}x^5e^{-x^3}\ln(1+x)\mathrm dx=\Gamma\left({1\over 3}\right)-\...
- 8,090
5
votes
2
answers
257
views
Convolution with Gaussian, without distribution theory, part 1
I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following:
Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and
$$
u(t,x)=...
- 1,354
4
votes
2
answers
3k
views
How to compute the integral $\int_{-\infty}^\infty e^{-x^2/2}\,dx$?
Yes, I know that this is very similar to $\int_{-\infty}^\infty e^{-x^2}\,dx$, which has been answered a million times, but I still don't know how to apply the technique from that integration to mine....
- 1,167
3
votes
0
answers
2k
views
Fourier transform of a truncated Gaussian function
I am trying to take a Fourier transform of a truncated Gaussian function $f(t,T)=e^{-at^2}I(|t|\leq T)$, where $I(B)$ is the indicator function returning one when boolean variable $B$ is true and zero ...
- 5,406
3
votes
2
answers
439
views
How can I evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar coordinates [duplicate]
I know from probability class that the area under the bell curve $e^{-x^2}$ is $\sqrt{\pi}$. I would like to be able to verify this, so in other words, solve this integral:
$$\int_{-\infty}^{\infty} e^...
- 4,334
3
votes
2
answers
192
views
Strange integrand: $\int \sqrt{x} e^{x}dx$
Is it possible, and if yes, how, to evaluate an integral like $\int \sqrt{x} e^{x}dx$? I have heard of the Gaussian function which integrates to $\sqrt{\pi}$ but what about this? Thank you.
- 33
3
votes
2
answers
347
views
Integral involving product of arctangent and Gaussian
During some research, it became desirable to compute
$$ \int_{\mathbb{R}^2} \arctan{ (y / x) } \exp{ [ - c ( (x - a)^2 + (y - b)^2 ) ]} d x dy $$
I am aware of the solution in the case when $a = b = 0$...
- 753
3
votes
1
answer
8k
views
Expected value of $X^n$ for normal distribution
Let there be a random variable $X$ and $N(0, \lambda ^2)$ given.
I am to find $E[X^n]$ knowing that $\Gamma \big(\frac{1}{2} \big) = \frac{\pi}{2}$.
I started with the definition:
$$E[X^n] = \frac{1}...
- 2,868
2
votes
1
answer
2k
views
Gaussian integral with error function
Given the integral:
$$\int_{0}^{+\infty}dz\, e^{-z^2} \text{erf}(z+\beta)=\frac{\sqrt{\pi}}{2}\left(1-\frac{1}{2}\text{erfc}^2\left(\frac{\beta}{\sqrt{2}}\right)\right),$$
where $\text{erf}(z)$ and $\...
- 235
2
votes
0
answers
91
views
An integral involving a Gaussian and a power of a normal cumulative distribution function
Being inspired by How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$? we formulated the question below.
Let $c \in (0,1/\sqrt{2})$ and let $n \in \Bbb{N}$. Then let $\phi(x) : =\frac{\...
- 11.4k
2
votes
1
answer
81
views
Convolution with Gaussian, without dstributioni theory, part 3
I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following:
Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and
$$
u(t,x)=...
- 1,354
1
vote
1
answer
50
views
How to prove $\sup_{t\in [a, b]} \int_{\mathbb R^d} |f_t (x)|^p \, \mathrm d x<\infty$ in higher dimension?
For $t>0$, we define
$$
g_t:\mathbb R^d \to \mathbb R_{\ge 0}, x \mapsto e^{- |x|^2 /t}.
$$
Let $t_n, t >0$ such that $t_n \to t$ as $n \to +\infty$. I have proved that
Lemma $g_{t_n} \to g_t$ ...
- 5,657
1
vote
1
answer
73
views
Convolution with Gaussian, without distribution theory, part 2
I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following:
Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and
$$
u(t,x)=...
- 1,354
1
vote
1
answer
327
views
Proof of a Gaussian Integral property
I'm working through some old integrals and I found one that's interesting. I can't quite remember how it's proved, so if someone could set me off in the right direction, it would be really helpful. ...
- 783
1
vote
1
answer
306
views
Gaussian Integral over matrix elements with correlation
$\mathbf{J}$ is a random matrix where $J_{ij}$ follows a Gaussian distribution.
Consider the following integral:
$$I=\int\left(\prod_{ij}\mathrm{d}J_{ij}\right) \exp\left\{-\frac{N}{2} \sum_{i, j, k}...
- 115
0
votes
2
answers
749
views
Derivation of the gaussian integral
Can someone explain what's going on between the second equals sign, where things are converted to $r$s?
- 185
0
votes
1
answer
236
views
Applying Fourier transform to a gaussian
Let $$G_\beta(w) = e^{\beta w^2}$$
Now I get the process of applying a fourier transform (or inverse) to get a new gaussian:
$$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the ...
- 5,619
0
votes
0
answers
38
views
If $f \neq 0$, do we have $\int_{ x_i \geq 0} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0$ for some $c \in (0,\infty)^n$?
Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a bounded measurable mapping. Next, define
\begin{equation}
\mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\}
\end{...
- 7,673
-1
votes
1
answer
79
views
Gaussian integral with complex linear component
I want to prove this gaussian integral is equivalent to left side using cauchy's theorem:
$$I=\int^\infty_{-\infty} dx \exp(-ax^2+bx)=\sqrt{\frac{\pi}{a}}e^{b^2/4a} $$
with $a \in\mathbb R$, $a>0$ ...
- 9
-1
votes
3
answers
363
views
Evaluating $\int_0^\infty x^2e^{-\alpha x^2}dx$ and $\int_0^\infty xe^{-\alpha x^2} dx$ knowing $\int_0^\infty e^{-\alpha x^2}dx$ [closed]
As the title, question 5 in this picture
https://i.stack.imgur.com/P52hf.jpg
thanks
- 47
12
votes
2
answers
359
views
Is $g(x)= \frac{x_1^2}{r_1^2}+\frac{x_2^2}{r_2^2}- c$ a unique solution of $E[g(Z)\mid M=\mu]=0, \forall \mu \in \text{ellipse } C$ for $Z$ Gaussian
Let $Z \in \mathbb{R}^2$ be an i.i.d. Gaussian vector with mean $M$ where $P_{Z\mid M}$ is its distribution.
Let $g: \mathbb{R}^2 \to \mathbb{R}$ and consider the following equation:
$$
E[g(Z)\mid M=\...
- 5,985
9
votes
1
answer
6k
views
What is the expected value of cosine of a multivariate Gaussian?
Suppose $X \sim \mathcal{N}\left(\mu, \Sigma\right)$. How do I evaluate $\operatorname{E}\left[\cos \left(t^{T}X \right) \right] $ and $\operatorname{E}\left[\sin \left(t^{T}X\right) \right] $? Does ...
- 993
9
votes
1
answer
204
views
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?
Is the following integral a convergent integral? Can we compute it, precisely?
$$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$
Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq \mathbb{R}^{n^...
- 848
8
votes
3
answers
956
views
Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$
I have to find
$$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$
I think we could use
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don't know how.
Thanks.
8
votes
4
answers
361
views
Finding the anti-derivative of $ \frac{e^{-c y^2 }}{y\sqrt{y^2-1}}$
I am trying to evaluate the integral
\begin{align}
\frac{1}{2\sqrt{2}\pi}\int_{0^{-}}^{t} ds \ \frac{e^{-x^2/2S^2(t,s) }}{\Sigma(s) S(t,s)}
\end{align}
where $S(t, s) = 2D(t-s)+\frac{\Sigma(s)}{2}$ ...
- 472
8
votes
7
answers
1k
views
How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$
How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$)
$$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$
Integration by parts doesn't seem to make it ...
- 4,552
8
votes
1
answer
164
views
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)
We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$
We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann ...
- 848
7
votes
1
answer
219
views
The integral $\int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^x)^2} dx$.
Let
$$T(n) = \int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^{x})^n} dx.$$
We have that
$$ T(0) = \sqrt{\pi} \text{ and } T(1) = \frac{\sqrt{\pi}}{2}$$
and also that
$$ T(3) = \tfrac{3}{2} T(2) - \frac{\...
- 145
6
votes
2
answers
1k
views
Integral involving erf and exponential
Problem
I would like to compute the integral:
\begin{align}
\int_{0}^{+\infty} \text{erf}(ax+b) \exp(-(cx+d)^2) dx \tag{1}
\end{align}
I have been looking at this popular table of integral of the ...
- 61