0
votes
1answer
21 views

Determining a value $c$ such that the function $f(c) = \displaystyle\sum_{i=1}^n \left|\frac{y_i-c}{y_i}\right|\times v_i$ is minimized

I'm trying to construct prediction model for a variable of interest, based on a set of input values. I have a set of validation data and their predictions (by my model) and now I need to asses whether ...
18
votes
1answer
309 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
45 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 ...
14
votes
2answers
147 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
130 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.
28
votes
3answers
434 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
2answers
99 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
1
vote
3answers
95 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
482 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 ...
17
votes
3answers
213 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 ...
0
votes
1answer
75 views

truncation error - help

I am trying to understand the concept of local truncation error and came accross this in my lecture notes: what I don't understand here is where the term 'O' comes from and what it stands for in ...
0
votes
0answers
66 views

Truncation error and difference method

I am stuck on the following question. I am not sure of how to calculate the truncation error for the difference method any help would be appreciated thank you!
5
votes
1answer
241 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
413 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: $$ \int_{0}^{1} e^{- x^{2}} dx = ~? $$