Use this tag for the error and complementary error functions (erf and erfc). These are special functions formed by taking definite integrals of the Gaussian/normal distribution function.

learn more… | top users | synonyms

2
votes
1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
1
vote
1answer
40 views

The so-called error function defined as: $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}erf(x)$ satisfies the differential equation: ...
0
votes
1answer
30 views

derivative of error function

How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial ...
0
votes
1answer
35 views

Find the 'rough' error bound to the composite simpson rule

Provide a rough error bound for the following composite simpsons rule. I am aware that the upper bound is $f$ to the forth derivative evaluated at some $t$ in the open interval ...
2
votes
2answers
92 views

Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei} $ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= ...
0
votes
0answers
14 views

Error function relation to the normal cumulative distribution function

A CDF for a normal standard is the following: $$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\phi^2/2} d\phi$$ I have the following relation in my notes which I am not very sure how they ...
0
votes
0answers
12 views

Total Error Bound for $D_1(h)$ Approximation

I am asked to find a total error bound for the approximation $D_1(h)$ in terms of $M_2=\max|f''(x)|$ and $\varepsilon=|f(x)-\bar{f}(x)|$. I began research what exactly $D_1(h)$ is and found ...
5
votes
1answer
104 views

Computing the inverse error function

I need a formula/series to compute the inverse error function, which is the inverse of $$ \operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{e}^{-t^2} \,\mathrm{d}t. $$ ...
2
votes
2answers
59 views

How to compute this integral without the use of the error function?

I was watching this: https://youtu.be/qQ-56b_LvOw?t=4484 And this integral came up. $$\int_{0}^{\infty}{(2x^2+1)e^{-x^2}}dx$$ To which the answer was $\sqrt{\pi}$. They made it clear that you ...
1
vote
1answer
41 views

Deriving Separate Forms of the Error Function

I noticed after evaluating a form of the error function $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$$ on WolframAlpha that another integral representation for $x\in \mathbb{R}$ is ...
3
votes
4answers
63 views

How to prove $P(x)=1-\frac{x^2}{2}$ is a good approximation of order $3$ for $f(x)=\cos x$ near $x=0$?

Let $f$ be a function and we want to approximate $f$ using a different function $P$ near $x=0$. The error of approximation is $E(x)=f(x)-P(x)$. If the approximation is going to be any good, we want ...
0
votes
1answer
35 views

$\int\limits_0^{10}e^{-0.04t}\cdot e^{-0.001t^2}dt$

I need to find the following integral $$\int\limits_0^{10}e^{-0.04t ~-0.001t^2}dt$$ This integral seems to "scream" for the error function, but I have never worked with the error function yet, so I ...
1
vote
3answers
28 views

differentiation of $\operatorname{erfc}(\sqrt{ax})$

I need your help to figure out the derivative of $\operatorname{erfc}(\sqrt{ax})$ with respect to $x$. Based on my knowledge on Wolfram references, they cite that: $$\frac{d ...
4
votes
3answers
151 views

Definite integral involving an error function

Let $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int\limits_{0}^x e^{-t^2}dt$$ be the error function. Then, I have tow questions. For a positive integer $n$, is there a close-form solution of ...
1
vote
1answer
34 views

Stable algorithm for computation of $\Phi(20)$, when $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$

Let $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$, i.e $\Phi$ is the MacLaurin series of the function $\displaystyle ...
0
votes
0answers
17 views

What are disadvantages of solving difference equations backwards?

Say you have the initial value of 100th and 99th term and you want to see how much error is in 1st term instead of picking values from the table for 1st and 2nd term and propagating to 99th and 100th ...
0
votes
0answers
27 views

integration of product of error function

Is there a more simple formulae for the following integral $$ \int_{a}^{+\infty} erf(\alpha x).erf(\beta x) \frac{1}{x^2} \:\mathrm{d}x $$ where $a>0$, $\beta>0$ and $\alpha>0$
3
votes
2answers
88 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
1
vote
2answers
42 views

Error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule.

Task is to define the exact error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule using n subintervals. I know the error term is $E(f)=\frac{1}{24}(b-a)f^{''}(\varepsilon)h^{2}$ but im ...
0
votes
0answers
17 views

continued fraction expansion of the complementary error function

Could someone please explain to a non-mathematician why the continued fraction expansion of the complementary error function is the following: fraction How does one come to this fraction as ...
1
vote
0answers
35 views

Integration involving Error function

I am interested in the following integral $$\int_0^\infty x^n\text{Erf}[ax]j_m(bx)dx,$$ where Erf is the error function $j_n$ is the spherical Bessel function of first kind. Does the analytical ...
0
votes
1answer
33 views

Solution for $g(x) - \int_0^y e^{t^2}\,{\rm d}t$.

Given the equation $g(x) - \int_0^y e^{t^2}\,{\rm d}t = 0$, with $g\colon \Bbb R \to \Bbb R$ of class ${\cal C}^\infty$, show that for each $x \in \Bbb R$ there is a unique $y = y(x)$ that solves ...
0
votes
0answers
61 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
0
votes
0answers
14 views

Minimizing error of estimation in a differential equation system

I have a system of equations which describe dynamic nature of a system. There is no closed form solution for this system. System of equations is as follow: $$I_c=I_1 + I_2 + I_3$$ $$R_3 = ...
1
vote
0answers
27 views

How are subtraction operations better conditioned than addition?

I was reading about error analysis in numerical methods, particularly about calculating the sum of arrays. I understand how naively summing all elements can lead to accumulation of error especially ...
2
votes
1answer
300 views

differential equation with $e^{x-t^2/2}$

I don't manage to solve the following DE $$y''(x)=\int_{-\infty}^{\frac{x^2}{2}} e^{x-\frac{t^2}{2}} \,\mathrm{d}t, \quad x > 0 , \quad y(0) = 0 , \quad y'(0) = 0 $$
2
votes
1answer
22 views

Proof that Normal Distribution is Normalized

How do we know that $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{(-x^2/2)}dx$ = 1. Or how do we know that the normal distribution is normalized? Or how do we know $erf(\infty) = 1$ ?
0
votes
2answers
52 views

Hint for integrating exp(x-x^2)

The function $e^{x-x^2}$ is zero if $x \to \infty$ or $x \to -\infty$ it looks like a normal-distribution-curve with the max. value at $x=0.5$. Has somebody a hint for integrating it from $-\infty$ ...
0
votes
1answer
54 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
1
vote
2answers
71 views

Integral of the error function

So I know that $$\displaystyle \int_{0}^{\infty} \text{erf}(x) dx$$ does not converge so I am assuming that $$\displaystyle \int_{0}^{\infty} \frac{\text{erf}(x)}{x} dx$$ does not converge? Is ...
0
votes
1answer
36 views

Inverse error function, its analytic continuation and Hardy space

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse ...
1
vote
0answers
37 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
2
votes
0answers
27 views

Solving Partial Differential Equation with Self-similar Solution

$$ Greetings, $$ So, I have a heat equation to be solved for in the form of $$ \frac{\partial f(x,t)}{\partial t} = \frac{\partial^2 f(x,t)}{\partial x^2} $$ for t = [0,+inf) and x = (-inf,+inf) and ...
1
vote
0answers
15 views

Show that this conversion between $\Phi$ and erf(z) holds for all z

I am trying to wrap my head around the connections between the standard Normal distribution and the error function. I could use some help working through the following problem. Show that the ...
2
votes
3answers
102 views

Integral of $e^{-x^{2}}$ and the error function

How to integrate $e^{-x^{2}}$? When I used geogebra I got the answer as $\frac{1}{2}\sqrt{\pi}\operatorname{erf}(x)$. What is $\operatorname{erf}(x)$ ? How to arrive at this answer?
0
votes
0answers
19 views

Integrating the product of two imaginary error functions

I want to evaluate the integral of a product of two imaginary error functions as in the following: $$\int^{L_2}_{-L_2} \operatorname{erfi}(g(y)) \operatorname{erfi}(h(y)) \, dy$$ where $g(y)$ and ...
3
votes
1answer
58 views

Prove $e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$

It seems to me that $$e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$$ This integral seems to converge for all $x\in\mathbb{C}$ I came upon this ...
0
votes
1answer
20 views

Modeling Gaussian components with standard vs exact functions

I'm studying a paper on modeling DNA histograms. It presents two alternative formulas for modeling Gaussian components: Standard form: $G(x) = ...
1
vote
1answer
69 views

Difference between the Error function and Normal distribution function?

I have just started reading about the Error function but Distribution theory is not my strong point. So I apologize in advance for asking really simple questions about it. So the Error function is ...
4
votes
0answers
48 views

How do I symbolically compute $\int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x$?

I want to symbolically write (in the form of a series), the integral of: $$ \int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x, \text{where }\{x, a, b\} \subset \mathbb{R} $$ The ...
0
votes
0answers
54 views

Convert an integral with exponential of cosine to the error function

I have an expression that looks like $$\frac{1}{T}\int_0^T\left(e^{\alpha_\theta(\cos\frac{\pi t}{T})^2}-1\right)\,dt,$$ where $\alpha_\theta,T$ are constants. I'd like to rewrite it into a form ...
0
votes
0answers
12 views

Taylor series expansion - error term

Consider the equation $$ f'(x) \approx \frac{1}{12h} \left( -f(x+2h) + 8f(x+h) - 8f(x-h) + f(x-2h) \right)$$ I wish to determine the $\textbf{error}$ term of this approximation (and I don't know how ...
1
vote
1answer
20 views

Where does mean and standard deviation go in the error function?

The error function is defined as $$ \textrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt~.$$ However, the normal distribution can take a more general form than the definition of the error ...
0
votes
0answers
150 views

Integration with Log of error function (erf)

Can anybody help me evaluating the closed-form or an approximate form of $H(x) = \int P(x) \ln(P(x)) \Bbb dx$ where $P(x) = \frac{C(x)}{v\int C(x) \Bbb dx}$ and $C(x) = {\frac ...
0
votes
0answers
42 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
2
votes
0answers
58 views

Solve:$\int_{0}^{t}{{\left(\cos({…})+\sin({…})\right)} \frac{\lambda^2 e^{(…)}}{\sqrt{\pi (t-r)}} \text{Erfc}{\left(… \right)} }~\mathrm{d}r$

I have another nasty integral to solve as follow: $$ I(t)=\int_{0}^{t}{{\left(\cos({\frac{\gamma}{4(t-r)}})+\sin({\frac{\gamma}{4(t-r)}})\right)} \frac{{\lambda^2} e^{2 \lambda^2 r+ \lambda ...
0
votes
0answers
57 views

Maximum Error Bound on a Cubic Spline, Chebyshev Polynomials

Most notations for the error resulting from interpolation of cubic splines (e.g. http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture12.pdf) requires knowledge of the original function to ...
0
votes
0answers
41 views

how to derive error function

I saw many places where erf(x) is used. As per wiki it is defined as gauss error function shown here . I recently used this to derive the cumulative probability density function for normal ...
0
votes
0answers
17 views

How can I calculate percent error with a denominator of 0? [duplicate]

I am using the MAPE formula, ABS(actual-forecast)/actual *100, but this raises problems when the actual values are zero. Is it possible to add 1 to the denominator and calculate this way? Would I get ...
0
votes
1answer
50 views

Asymptoting to 0: is erfc(z) quicker than exp(-z)?

If I have a function of the form $\mathrm{erfc}\left(z\right)/e^{-z}$, should I expect its limit at large $z$ to be $0$ or $\infty$? My instinct is that it should be $0$, by considering the ...