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Chapter 11: Q10.5P (page 552)

(a) Express E₁(x)as an incomplete $螕$ function.
(b) Find the asymptotic series for E₁(x).

Short Answer

Expert verified

The required expression is, $E_{1} (x) = {鈭�}_{x}^{鈭�} t^{- 1} e^{- t} d t = 螕 (0, x)$ .
The required series is, $E_{1} (x) = \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .]$ .

Step by step solution

01

Given information

E₁(x) and gamma function is given.

02

Definition of a Gamma function

The Gamma Function is defined as

$螕 (p) = {鈭�}_{0}^{鈭�} x^{p - 1} e^{- x} d x, p > 0$

03

Begin with the definition of a Gamma function

(a)

Write the Gamma function.

$螕 (p) = {鈭�}_{0}^{鈭�} x^{p - 1} e^{- x} d x, p > 0$

Substitute p = 0.

$\begin{array}{rcl} 螕 (0, x) & = & {鈭�}_{0}^{鈭�} t^{- 1} e^{- x} d \\ = & E_{1} (x) \end{array}$

04

Write the expansion of an incomplete Gamma function.

(b)

Expand the incomplete Gamma function.

$螕 (p, x) = x^{p - 1} e^{- x} [1 + \frac{(p - 1)}{x} + \frac{(p - 1) (p - 2)}{x^{2}} + \frac{(p - 1) (p - 2) (p - 3)}{x^{3}} + . . .]$

Substitute p = 0.

$\begin{array}{rcl} 螕 (0, x) & = & E_{1} (x) \\ = & \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .] \end{array}$

Hence, the asymptotic series is $\begin{array}{rcl} \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .] \end{array}$ .

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Most popular questions from this chapter

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

12. ${鈭�}_{0}^{\frac{1}{2}} \frac{\sqrt{100 - t^{2}}}{\sqrt{1 - t^{2}}} d t$

Show that for $k = 0$ , $u = F (蠒, 0) = 蠒, s n u = s i n u, c n u = c o s u, d n u = 1 .$ and for $k = 1$ $u = F (蠒, 1) = l n (s e c 蠒 + t a n 蠒), 蠒 = g d u, s n u = t a n h u, c n u = d n u = sech u$ .

Use Stirling鈥檚 formula to evaluate $\underset{n 鈫� 鈭�}{l i m} \frac{{(n!)}^{\frac{1}{n}}}{n}$ .

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

Express the complementary error function erfc(x)as an incomplete $螕'$ function (see Problem 2) and use your result in Problem 2to obtain (again) the asymptotic expansion oferfc(x) as in (10.4) .

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