0
votes
1answer
19 views

Where does the “2” come from in deriving Normal PDF from its kernel?

I'm trying to train myself to recognize probability densities by deriving PDFs from bare kernel functions. In other words, I find a constant expression by integrating a kernel function over its ...
1
vote
1answer
15 views

Integrating the error function in a calculation related to Brownian motion

I wish to calculate the probability that a standard linear Brownian motion $B(t)$, $t\ge 0$, will be at time $t_0$ inside the interval $[a,b]$, and at time $t_1$ in the interval $[c,\infty)$. To do ...
2
votes
0answers
28 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
2
votes
2answers
81 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
1
vote
1answer
27 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
1
vote
1answer
39 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
2
votes
3answers
86 views

Integrate $ x\cdot e^{i\omega x - x^2}$ from $0$ to $\infty$

Can anybody help me in solving the following integral: $$\int_0^\infty x\cdot e^{i \omega x-x^2}\,dx\quad?$$ Any help/hints will be highly appreciated. Thanks
18
votes
1answer
326 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
0
votes
1answer
47 views

Changing the limits of integration from $-\infty$ to $\infty$

Is it fair to change from $$ \int_{-\infty}^{a} \exp \left( -t^2 \right) \mathrm dt $$ replacing $t$ with $-t$ $$ \int_{-a}^{\infty} \exp \left( -t^2 \right) \mathrm dt $$ and thus gaining the ...
15
votes
2answers
272 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
4
votes
2answers
142 views

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
30
votes
3answers
504 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ but have problems ...
1
vote
1answer
200 views

How to derive integrals with error function?

How to derive this integral $\int_{-\infty}^{\infty}erf(\lambda x)\mathcal{N}(\mu, \sigma ^2)dx$ and this $\int_{-\infty}^{\infty}(erf(\lambda x)-const)^2\mathcal{N}(\mu, \sigma ^2)dx$ where ...
11
votes
1answer
176 views

Integral $\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx$

Consider the following integral: $$\mathcal{A}=\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx,\tag1$$ where $\operatorname{erfi}(x)$ denotes the imaginary error ...
2
votes
1answer
48 views

How to show integral is related to complementary error function?

I have to somehow show that: $$ \int_0^\infty{\frac{e^{-a t}}{t^{1/2}(t+x)}} \textrm{d}t=\frac{\pi}{\sqrt{x}} e^{a x} \textrm{erfc}(\sqrt{ax}) $$ I've tried substituting $u=\sqrt{t}$ to get $$ 2 ...
1
vote
3answers
101 views

Help with exponential integrals

I'm trying to find a nice expression for the following function \begin{equation} f_k(x)=\int_0^\infty y^k (x+y) e^{-(x+y)^2} \text{d}y. \end{equation} So far I know that \begin{equation} ...
20
votes
2answers
566 views

Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$

Is it possible to find a closed form (possibly using known special functions) for this integral? $$\int_0^\infty\left(5\,x^5+x\right)\operatorname{erfc}\left(x^5+x\right)\,dx$$ where ...
1
vote
1answer
84 views

Proof of an integral identity

I would appreciate it if you could help proving the following identity which are two forms of complimentary error function $$\frac{2}{\sqrt{\pi}}\int_z^\infty e^{-t^2} ...
17
votes
3answers
226 views

A closed form for $\int_0^\infty e^{-a\,x} \operatorname{erfi}(\sqrt{x})^3\ dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the following parameterized integral $$I(a)=\int_0^\infty ...
16
votes
2answers
308 views

A closed form for $\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\ e^{-x\sqrt{2}}}{x}dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the integral $$I=\int_0^\infty\frac{\sin(x)\ ...
16
votes
1answer
342 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
2
votes
1answer
99 views

Evaluating $\int\frac{e^{-x^2}}{(1+2x^2)^2}dx$

I'm trying to evaluate the following indefinite integral: $$ \int\frac{e^{-x^2}}{(1+2x^2)^2}dx $$ According to Wolfram|Alpha, this integral evaluates to: $$ \int \frac{e^{-x^2}}{(1+2 x^2)^2} dx = ...
1
vote
1answer
176 views

Rewriting the integral $\mathrm{erf}(x) = \frac{1}{\sqrt{\pi}} \int_{-x}^x e^{-t^2} dt.$

I'm trying to implement an equation into a programming language which doesn't have functions for integrals. However as it's many years since I've had any math exercise I'm having some trouble ...
3
votes
2answers
91 views

Advice on an integral involving the error function

I'd like to calculate the following integral: $$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{1+x}} - \frac{\sqrt{1+x}}{\beta}\right) \exp\left(-\frac{x}{\gamma}\right)\, dx,$$ where ...
1
vote
0answers
258 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} ...
1
vote
2answers
453 views

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ ...
5
votes
1answer
280 views

Integrating a product of exponential and complementary error function with square-root of variable in the denominator

I need to evaluate \begin{equation} \int_a^\infty \mathrm{erfc}\left( \frac{b}{\sqrt{c\cdot h}} \right) e^{-d\cdot h} dh \end{equation} where $\mathrm{erfc}(s) = \frac{2}{\sqrt{\pi}} ...
4
votes
3answers
430 views

How to compute $ \int_0^1 {e^{-x^2}} dx$

I know that $$ \int_{0}^{+ \infty} e^{- x^{2}} dx = \frac{\sqrt{\pi}}{2}. $$ My question is: $$ \int_{0}^{1} e^{- x^{2}} dx = ~? $$