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.

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Solving $\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right)$

I got as far as: $$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} ...
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1answer
31 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
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1answer
14 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
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1answer
70 views

How can I solve the integral in the error function $\mbox{erf}(x)$?

To get from this To this series I can't seem find the step-by-step solution anywhere.
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8 views

Difference Between Three Similar Error Reducing Algorithms

I found a Least Square Error Recognition algorithm that finds the least mean square error from a 2-d matrix element by element. Logistic regression from this site, on the other hand, seeks to ...
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1k views

Integral involving an error function

For $\sigma>0$, how can we prove that $$\frac{1}{2}\int_{-1}^1 \text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right) \, \mathrm{d}x= \text{erf}\left(\frac{\sigma}{2}\right)$$ where erf ...
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2answers
83 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll ...
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90 views

Convincing that $\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n-2}(2n+1)(n!)} \ne \pi$

A friend of mine in high school challenged me to calculate the value of the sum $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n-2}(2n+1)(n!)}$$ And then claimed that the answer was $\pi$ . But when I ...
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17 views

Why is the Riemann integral for the sum-of-squares error function multiplied by the joint density function of the underlying distribution?

In Bishop's Neural Networks for Pattern recognition , he expresses an error function $$ E= \frac{1}{2}\sum_{n=1}^{N}\sum_{k}\{(y_k(\mathbf{x}^{n};\mathbf{w})-t^{n}_{k}\}^2$$ where $\mathbf{x}^{n}$ ...
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56 views

two dimensional Gaussian integral with complex exponent of an absolute value

I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ...
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41 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
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11 views

How to solve this statistics problem? Error function

A store received 1500 glass bottles. The probability that the randomly chosen product is defective is 0.3% . Create the distribution law of the random variable X number of defective products , ...
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2answers
26 views

Error in ratio of two numbers

How tow find error associated with a ratio $R$, when both the numerator and denominator contains error. For example $$ R=\frac{(A \pm \Delta A)}{(B\pm\Delta B)}.$$
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2answers
65 views

Erf squared approximation

I found a nice and useful approximation of squared error function $$ \mathrm{erf}^{2}\!\left(x\right)=1-\exp\!\left(-\frac{\pi^{2}}{8}x^{2}\right)+\varepsilon\!\left(x\right). $$ I checked ...
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3answers
171 views

What am i doing wrong when solving this differential equation

$$ f(x) = \frac{\frac{d^2}{dx^2}[e^y]}{\frac{d}{dx}[e^y]} $$ Given that $f(x) = cx$ $$ \frac{c}{2}x^2 + k_1 = \ln(e^y y') $$ $$ k_2\int e^{\frac{c}{2}x^2} dx = e^y $$ $$ y = \ln(k_2\int ...
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1answer
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Integral involving the error function of log(x)

Looking for a closed form for the integral $$\int_0^{\infty } e^{-\left(\frac{a-\log (x)}{b}\right)^2} \left(\frac{1}{2} \text{erf}\left(\frac{a-\log (x)}{b}\right)+\frac{1}{2}\right) \, ...
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1answer
58 views

Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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37 views

How to compute an integral involving the error function

Solving a problem with one dimensional diffusion the following identity naturally arises $$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm ...
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44 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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40 views

Proof that $\int_0^\infty t^\gamma e^{t/2} \operatorname{erfc}(\sqrt{t}\,) \,dt$ leads to hypergeometric function

I'm looking for the proof of $$\DeclareMathOperator\erfc{erfc} \int_0^\infty t^\gamma e^{t/2} \erfc(\sqrt{t}\,) \, dt = \frac{2^{-2 \gamma -1} \Gamma (2 \gamma +2) \, _2F_1\left(\gamma +1,\gamma ...
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1answer
21 views

What can I imply from $\epsilon^\prime(x) \le -x\epsilon(x)$?

Imagine, that I can prove $$\epsilon^\prime(x) \le -x\epsilon(x)$$ for a function $\epsilon:\mathbb R \rightarrow \mathbb R_0^{+}$. Does this imply $$\epsilon(x) \le c e^{-\frac{x^2}{2}}$$ whereby $c ...
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1answer
31 views
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46 views

Integral of a gaussian over a slice of the plane

I need to evaluate the following $n$ real integrals: $$\int_{\frac{\pi}{2}-\frac{\pi}{n}}^{\frac{\pi}{2}+\frac{\pi}{n}}\int_0^\infty\frac{1}{\pi \sigma^2}e^{-\frac{|re^{i\theta}-i|^2}{\sigma^2}} \ dr ...
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2answers
24 views

Why is the Wronskian of these two functions equal to $\frac{2}{\sqrt{\pi}}$

I can't seem to get the right answer $ f = e^{\frac{y^2}{2}}$ and $g =e^{\frac{y^2}{2}}erf(y)$ where $erf(y) = \frac{2}{\sqrt\pi}\int_{0}^{y}e^{-\alpha^2}d\alpha$ I get $W = \frac{2}{\sqrt\pi} - ...
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2answers
84 views

How to integrate $e^{-x^2 -y^2}$?

How do you compute the following integral:$e^{-x^2 -y^2}$? Wolfram already gives the answer as: $\frac{1}{2} \sqrt{\pi} e^{-y^2} \text{erf}(x) + C$, but I have no idea how to get there. I tried ...
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2answers
52 views

probability: When $X\sim\mathcal{N}(\mu, \sigma^2)$, the med = $\mu$

The median of a PDF is defined as the point $x = $med for which $P[X\leq\text{med}] = 1/2$. Prove that if $X\sim\mathcal{N}(\mu, \sigma^2)$, then med $= \mu$. Since $P[X\leq\text{med}] = 1/2$, ...
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92 views

error function inverse limits

$$ \lim_{x \to 0+} e^{-(\operatorname{erfc}^{-1}(2x) - 2\operatorname{erfc}^{-1}(x))^2}(e^{\operatorname{erfc}^{-1}(x)^2} - e^{\operatorname{erfc}^{-1}(2x)^2}) $$ Where $\operatorname{erfc}(x) = ...
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1answer
23 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
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1answer
66 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
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1answer
28 views

Error function - Not seeming to come out right

I have reached two integrals: $$\int_{(x+L)/(2c\sqrt{t})}^\infty e^{-z^2} dz + \int_{-\infty}^{(x-L)/(2c\sqrt{t})} e^{-z^2} dz$$ Now the first evaluates just fine to ...
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98 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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1answer
61 views

Integral involving the error function

Is there a closed form solution to the integrals \begin{align} I_{c} &= \int_{0}^{\infty} \cos(a x) \, \operatorname{erf}(b x) \, dx \\ I_{s} &= \int_{0}^{\infty} \sin(a x) \, ...
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2answers
24 views

Behaviour of the error function as $z \rightarrow -\infty$?

I'm trying to find the behaviour of the error function, $erf(z)$ as $z \rightarrow -\infty$ $$erf(z) = \frac{2}{\sqrt{\pi}}\int_0^{z} e^{-s^2}\mathrm{ds}$$ I know that we can find the limit of ...
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1answer
49 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...
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1answer
127 views

Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
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49 views

Integral involving complimentary error function.

Essentially, I am trying to work out an integral of the form: $\int_{0}^{\pi }{x\cdot a\cdot \cos \left( x \right)e^{-\left[ a\cdot \sin \left( x \right) \right]^{2}}\mbox{erfc}\left[ -a\cdot \cos ...
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48 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
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1answer
181 views

Is there an analytical solution to Gaussian integral $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx$?

I wonder if there is an analytical solution to $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$ where $a, b>0$. I know, of course, that the antiderivative of the fraction is a version ...
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1answer
29 views

Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
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1answer
24 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 ...
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1answer
37 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 ...
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8 views

To find error in time histogram

I have a data which is recorded from a detector. Whenever the detector produce signal it records the time. I have recorded the data for several cycles, one cycle is 0 to 1 second. Finally I made the ...
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70 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: $$ ...
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1answer
62 views

Discrete Function Approximation Error - Which type? (Applied math, signals)

I have two functions, one derived via software, and we can call it the exact function, $f_{exact}$. The other is a result I got through hardware, and we can call it the approximation, $f_{approx}$. ...
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1answer
56 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
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121 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 ...
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2answers
87 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
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1answer
65 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
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1answer
41 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 ...
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1answer
64 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. ...