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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.

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Error Function -

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# Error Function

## Error Function Calculator

 Function ERF() ERFC() (Complement) ERF^-1() (Inverse) ERFC^-1() (Inverse Complement)

### 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 $x$.

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