Question

Replace x by ix in (9.1) and let t = iuto show that erf(ix) = ierfi(x), where erfi(x) is defined in (9.7).

Short answer

The expressions are proved below.

Step-by-step solution

Given information.

Error function is given.

Definition of an error function.

The error function (also known as the Gauss error function) is a complicated function of a complex variable that is defined as follows:

Error function$\mathit{\beta }\left(p,q\right)\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{{\mathbf{t}}^{\mathbf{p}\mathbf{-}\mathbf{1}}}{{\left(1+t\right)}^{\mathbf{p}\mathbf{+}\mathbf{q}}}\mathit{d}\mathit{t}$.

Begin with the definition of an error function.

Write the definition of an error function.

$\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{ix}{\int }}{e}^{-u^2}du$

Substitute the following in the integral.

$\begin{array}{rcl}s& =& iu\\ ds& =& idu\end{array}$

Substitute the values in the error function and continue evaluating the integral.

$\begin{array}{rcl}\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{ix}{\int }}{e}^{-u^2}du& =& \frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{x}{\int }}{e}^{s^2}\left(i\right)du\\ & =& i\frac{2}{\sqrt{\mathrm{\pi }}}\underset{0}{\overset{x}{\int }}{e}^{s^2}du\end{array}$

Hence, proved