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Chapter 11: Q11.6P (page 554)

Use equations (3.4) and (11.5) to show that $螕 (p) 鈻� p^{p} e^{- p} \sqrt{\frac{2 蟺}{p}} (1 + \frac{1}{12 p} + . . .)$ .

Short Answer

Expert verified

The equation has been proved.

Step by step solution

01

Given Information

The statement is $螕 (p) 鈻� p^{p} e^{- p} \sqrt{\frac{2 蟺}{p}} (1 + \frac{1}{12 p} + . . .)$ .

02

Definition ofthe sterling’s formula.

Sterling鈥檚 formula is used to simplify formulas involving factorial.

$n! 鈻� n^{n} e^{- n} \sqrt{2 蟺苍}$ .

03

Prove the statement.

The statement is $螕 (p) 鈻� p^{p} e^{- p} \sqrt{\frac{2 蟺}{p}} (1 + \frac{1}{12 p} + . . .)$ .

The sterling鈥檚 formula is role="math" localid="1664882157502" $螕 n 鈻� n^{n} e^{- n} \sqrt{2 蟺苍}$ .

The equation (11.5) is expressed as,

$螕 (n + 1) 鈻� n^{n} e^{- n} \sqrt{2 蟺苍} (1 + \frac{1}{12 n} + \frac{1}{288 n^{2}} + . . .)$

The equation (3.4) is expressed as,

$螕 (n + 1) = n 螕 (n)$

The value of $螕 (n)$ becomes as follows.

$螕 (n) = \frac{1}{n} 螕 (n + 1) 螕 (n) = n^{n} e^{- n} \sqrt{2 蟺苍} (1 + \frac{1}{12 n} + \frac{1}{288 n^{2}} + . . .)$

Hence, the statement has been proven.

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Most popular questions from this chapter

The integral ${鈭�}_{0}^{鈭�} t^{p - 1} e^{- t} d t = 螕' (p, x)$ is called an incomplete $螕'$ function. [Note that if x = 0, this integral is $螕' (p)$ .] By repeated integration by parts, find several terms of the asymptotic series for $螕' (p, x)$ .

In Chapter 1, equations (13.5) and (13.6), we defined the binomial coefficients $(p n)$ whereis a non-negative integer butmay be negative or fractional. Show that $(p n)$ can be written in terms of $螕$ functions as

role="math" localid="1664340642097" $(\frac{p}{n}) = \frac{螕 (p + 1)}{n! 螕 (p - n + 1)}$

If $u = l n (s e c 蠁 + t a n 蠁)$ , then 蠁 is a function of u called the Gudermannian of u, $蠁 = g d u$ . Prove that: .

$u = l n t a n (\frac{蟺}{4} + \frac{蠒}{2}), t a n g d u = s i n h u, s i n g d u = t a n h u, \frac{d}{du} g d u = sech u$

Computer plot graphs of K(k) and E(k)in (12.3)for k = 0 - 1. Also plot 3Dgraphs of $F (蠒, k)$ and $E (蠒, k)$ in (12.1) for k = 0-1and $蠒$ from $0 - \frac{蟺}{2}$ and also from $0 - 2 蟺$ . Warning: Be sure you understand the notation used by your computer program; see text discussion just after (12.3) and Example 1.

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

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