Question

Prove that erf(x) is an odd function of x. Hint: Put t = -s in (9.1) .

Short answer

It has been proved that the erf(x) is an off function of x .

Step-by-step solution

Definition of error of function

The error of the function is defined as $\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\sqrt{\mathbf{\pi }}}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}{\mathit{e}}^{\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\mathit{d}\mathit{t}$.

Prove that error is odd

The value of integration is $erf\left(\mathrm{x}\right)=\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{\mathrm{x}}{\int }}{e}^{-{\mathrm{t}}^2}dt$.

Substitute -x in place of x .

$erf(-\mathrm{x})=\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{-\mathrm{x}}{\int }}{e}^{-{\mathrm{t}}^2}dt$

Let u = -s , then du = -ds .

The equation becomes as follows:

$$\begin{gathered} erf(-\mathrm{x})=\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{\mathrm{x}}{\int }}{e}^{-s^2}\left(-ds\right) \\ erf(-\mathrm{x})=-erf\left(\mathrm{x}\right) \end{gathered}$$

Hence, the statement has been proved.