Solution Found!

Physics / An Introduction to Thermal Physics 1 / Chapter B / Problem 17P

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

Change Question

Derive the general integration formulas B.36.

Chapter , Problem 17P

(choose chapter or problem)

Get Unlimited Answers

QUESTION:

Questions & Answers

QUESTION:

Derive the general integration formulas B.36.

ANSWER:

Solution 17P

The integration formulas required to be proved are required to be

$\int\limits_{0}^{\infty{}}\frac{{x}^{n}}{{e}^{x}-1}dx=\Gamma{}(n+1)\zeta{}(n+1)$

$\int\limits_{0}^{\infty{}}\frac{{x}^{n}}{{e}^{x}+1}dx=(1-\frac{1}{{2}^{n}})\Gamma{}(n+1)\zeta{}(n+1)$

It is already proved that ,

$\int\limits_{0}^{\infty{}}\frac{x}{{e}^{x}-1}dx=\zeta{}(2)$

$\int\limits_{0}^{\infty{}}\frac{x}{{e}^{x}+1}dx=\frac{1}{2}\zeta{}(2)$

The Riemann zeta function is defined as,

$\zeta{}(n)=1+\frac{1}{2{\ }^{n}}+\frac{1}{{3}^{n}}+.....=\sum\limits_{k=1}^{\infty{}}\frac{1}{k{\ }^{n}}$

Step 1

$\int\limits_{0}^{\infty{}}\frac{{x}^{n}}{{e}^{x}-1}dx=\int\limits_{0}^{\infty{}}{x}^{n}({e}^{x}-1{)}^{-1}dx=\int\limits_{0}^{\infty{}}{x}^{n}({e}^{-x}dx+{e}^{-2x}dx+{e}^{-3x}dx+.....)$

$=\int\limits_{0}^{\infty{}}{x}^{n}{e}^{-x}dx+\int\limits_{0}^{\infty{}}{x}^{n}{e}^{-2x}dx+\int\limits_{0}^{\infty{}}{x}^{n}{e}^{-3x}dx+.....$

$=n!+n!\frac{1}{{2}^{n+1}}+n!\frac{1}{{3}^{n+1}}+n!\frac{1}{{4}^{n+1}}+.....$

$=n![1+\frac{1}{{2}^{n+1}}+\frac{1}{{3}^{n+1}}+\frac{1}{{4}^{n+1}}+.....]$

$=\Gamma{}(n+1)\zeta{}(n+1)$

Since, $\Gamma{}(n+1)=n!$

$\zeta{}(n+1)=1+\frac{1}{2{\ }^{n+1}}+\frac{1}{{3}^{n+1}}+.....=\sum\limits_{k=1}^{\infty{}}\frac{1}{k{\ }^{n+1}}$

Therefore, $\int\limits_{0}^{\infty{}}\frac{{x}^{n}}{{e}^{x}-1}dx=\Gamma{}(n+1)\zeta{}(n+1)$

Add to cart

OpenIntro Statistics 3
Family size. Suppose we want to estimate household size, where a household is defined as people living together in the same dwelling, and sharing living accommodations. If we select students at random at an elementary school and ask them what their family size is, will this be a good measure of house

Solution Found!

Derive the general integration formulas B.36.

Questions & Answers

Study Tools You Might Need

Chapter: 2 Problem: 4

Chapter: Problem: 4

Chapter: Problem: 1.24

Chapter: 5 Problem: 46

Chapter: 15 Problem: 17

Chapter: 3 Problem: 9

Chapter: 4 Problem: 5

Chapter: 13 Problem: 1

Not The Solution You Need? Search for Your Answer Here:

Login

Register

Solution Found!

Derive the general integration formulas B.36.

Questions & Answers

Study Tools You Might Need

Chapter: 2 Problem: 4

Chapter: Problem: 4

Chapter: Problem: 1.24

Chapter: 5 Problem: 46

Chapter: 15 Problem: 17

Chapter: 3 Problem: 9

Chapter: 4 Problem: 5

Chapter: 13 Problem: 1

Not The Solution You Need? Search for Your Answer Here:

Login

Register

Reset password