Search for a tool
Error Function

Tool to evaluate the value of the error function noted erf() or its complementary erfc(); special functions used in probability, statistics or physics.

Results

Error Function -

Tag(s) : Functions

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Error Function' tool for free! Thank you!

Error Function

Error Function Calculator






See also: Calculator

Answers to Questions (FAQ)

What is the error function? (Definition)

The error function (sometimes called Gaussian error function) is denoted $ \operatorname{erf} $ and is defined by the formula $$ \operatorname{erf}(x)= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, \mathrm{d}t $$

The error function returns a result between -1 and 1.

Example: $ \operatorname{erf}(0) = 0 $, $ \operatorname{erf}(-\infty) = -1 $, $ \operatorname{erf}(+\infty) = 1 $

The erf function is an odd function

What is the complementary error function? (Definition)

The complementary error function is defined by $$ \operatorname{erfc}(x) = 1-\operatorname{erf}(x) $$

Example: $ \operatorname{erf}(x) + \operatorname{erfc}(x) = 1 $

What is the inverse error function? (Definition)

The inverse error function is defined by the inverse (reciprocal) function $$ \operatorname{erf}^{-1}(\operatorname{erf}(x)) = x $$

Example: Knowing $ \operatorname{erf}(x) = 0.1 $ then it is possible to find $ x = \operatorname{erf}^{-1}(0.1) \approx 0.0088625 $

What is the integer series corresponding to the error function?

The expansion in integer series of 'erf ()' allow a fast computation (with approximations) by the formula: $$ \operatorname{erf}(x)= \frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{n!(2n+1)} = \frac{2}{\sqrt{\pi}} \left( x - \frac{z^3}{3} + \frac{z^5}{10} - \frac{z^7}{42} + \frac{z^9}{216} + \cdots \right) $$

The inverse function has a series expansion: $$ \operatorname{erf}^{-1}(x) = \frac{1}{2} \sqrt{\pi} \left( x + \frac{\pi}{12}x^3 + \frac{7\pi^2}{480} x^5 + \frac{127\pi^3}{40320} x^7 + \frac{4369\pi^4}{5806080} x^9 + O(x^{10}) \right) $$

What are error function properties?

The image of a negative value $ -z $ is the opposite of the image of the positive value: $ \operatorname{erf}(-z) = -\operatorname{erf}(z) $ (symmetry)

The derivative of the error function is $$ \frac{d}{dz} \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}e^{-z^2} $$

The integral of the error function is $$ \int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi}}+x \operatorname{erf}(x) + c $$

The error function is bounded between -1 and 1, that is, $ -1 \leq \operatorname{erf}(x) \leq 1 $ for all real $ x $.

Source code

dCode retains ownership of the "Error Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Error Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Error Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Error Function" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cita dCode

The copy-paste of the page "Error Function" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Error Function on dCode.fr [online website], retrieved on 2024-03-29, https://www.dcode.fr/error-function

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Error Function' tool for free! Thank you!


https://www.dcode.fr/error-function
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.
 
Feedback