Chapter 11: Q11.6P (page 554)

Use equations (3.4) and (11.5) to show that $Γ (p) □ p^{p} e^{- p} \sqrt{\frac{2 π}{p}} (1 + \frac{1}{12 p} + . . .)$ .

Short Answer

Expert verified

The equation has been proved.

Step by step solution

Given Information

The statement is $Γ (p) □ p^{p} e^{- p} \sqrt{\frac{2 π}{p}} (1 + \frac{1}{12 p} + . . .)$ .

Definition ofthe sterling’s formula.

Sterling’s formula is used to simplify formulas involving factorial.

$n! □ n^{n} e^{- n} \sqrt{2 πn}$ .

Prove the statement.

The statement is $Γ (p) □ p^{p} e^{- p} \sqrt{\frac{2 π}{p}} (1 + \frac{1}{12 p} + . . .)$ .

The sterling’s formula is role="math" localid="1664882157502" $Γ n □ n^{n} e^{- n} \sqrt{2 πn}$ .

The equation (11.5) is expressed as,

$Γ (n + 1) □ n^{n} e^{- n} \sqrt{2 πn} (1 + \frac{1}{12 n} + \frac{1}{288 n^{2}} + . . .)$

The equation (3.4) is expressed as,

$Γ (n + 1) = n Γ (n)$

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

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

Hence, the statement has been proven.

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Use equations (3.4) and (11.5) to show that $Γ (p) □ p^{p} e^{- p} \sqrt{\frac{2 π}{p}} (1 + \frac{1}{12 p} + . . .)$ .

Short Answer

Step by step solution

Given Information

Definition ofthe sterling’s formula.

Prove the statement.

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Use equations (3.4) and (11.5) to show that Γ(p)□ppe-p2πp(1+112p+...).

Short Answer

Step by step solution

Given Information

Definition ofthe sterling’s formula.

Prove the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

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Study anywhere. Anytime. Across all devices.

Use equations (3.4) and (11.5) to show that $Γ (p) □ p^{p} e^{- p} \sqrt{\frac{2 π}{p}} (1 + \frac{1}{12 p} + . . .)$ .