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Chapter 11: Q5P (page 548)

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

Expert verified

The expressions are proved below.

Step by step solution

01

Given information.

Error function is given.

02

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 $β (p, q) = \int_{0}^{\infty} \frac{t^{p - 1}}{{(1 + t)}^{p + q}} d t$ .

03

Begin with the definition of an error function.

Write the definition of an error function.

$\frac{2}{\sqrt{π}} \int_{0}^{i x} e^{- u^{2}} d u$

Substitute the following in the integral.

$\begin{array}{rcl} s & = & i u \\ d s & = & i d u \end{array}$

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

$\begin{array}{rcl} \frac{2}{\sqrt{π}} \int_{0}^{i x} e^{- u^{2}} d u & = & \frac{2}{\sqrt{π}} \int_{0}^{x} e^{s^{2}} (i) d u \\ = & i \frac{2}{\sqrt{π}} \int_{0}^{x} e^{s^{2}} d u \end{array}$

Hence, proved

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Most popular questions from this chapter

Use Stirling’s formula to evaluate $\underset{n \to \infty}{l i m} \frac{{(n!)}^{\frac{1}{n}}}{n}$ .

The function $ψ (p) = \frac{d}{d p} I n Γ (p)$ is called the digamma function, and the polygama functions are defined by $ψ^{n} (p) = \frac{d^{n}}{d p^{n}} ψ (p)$ . [Warning: Some authors define $ψ (p)$ as $\frac{d}{d p} I n (p!) = \frac{d}{d p} I n Γ (p + 1)$ ).]

(a) Show that $ψ (p + 1) = ψ (p) + \frac{1}{p}$ . Hint: See (3.4).

(b) Use Problem 6 to obtain $ψ (p) □ I n (p) - \frac{1}{2 p} - \frac{1}{12 p^{2}} . . .$ .

Use a graph of $s i n^{2} θ$ and the text discussion just before (12.4)to verify the equations (12.4). Note that the area under the $s i n^{2} θ$ graph from $0 - \frac{π}{2}$ and the area from $\frac{π}{2} - π$ are mirror images of each other, and this will be true also for any function of $\sin^{2} θ$ .

(a) Express E₁(x)as an incomplete $Γ$ function.

(b) Find the asymptotic series for E₁(x).

Using (5.3) with (3.4) and (4.1), find $Γ (3 / 2)$ , $Γ (- 1 / 2)$ , and $Γ (- 3 / 2)$ in terms of $\sqrt{π}$ .

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