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Evaluate (Hint: Change variables to convert the integrand

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 8P Chapter B

An Introduction to Thermal Physics | 1st Edition

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

An Introduction to Thermal Physics | 1st Edition

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Problem 8P

Evaluate  (Hint: Change variables to convert the integrand to a Gaussian.) Then use the recursion formula to evaluate

Step-by-Step Solution:

To evaluate

Step 1:

        For the positive values of , the Gamma function  = ----(1)

        Evaluating for =

Step 2:

        Substituting =  in (1)

        = ---(2)

        = 2

        We know that = 1, therefore the above equation become

        = 2-----(3)

Step 3:

        From the Gaussian integral, using the result of Gaussian integral

        =

        Splitting the limits of the integral, we get

         =  +  =

        Each part is equal and the value for one part is

        = -----(4)

Step 4:

        Applying the result of (4) in (3)

        We get

        = 2()

        =  = 1.7725---(5)

Using Recursion formula

To evaluate

Step 1:        

        For any positive value of , the gamma function becomes

         = (-1)        ----(6)

        Substituting =

        = (-1)

        =         -----(7)

Step 2:

        We know that =

        =

        = = 0.8862        ----(8)

To evaluate

Step 1:

        The previous formula is used only for positive values of . When < 0 the gamma function becomes

         = -----(9)

         =

Step 2 of 2

Chapter B, Problem 8P is Solved
Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

This full solution covers the following key subjects: evaluate, hint, change, formula, gaussian. This expansive textbook survival guide covers 10 chapters, and 454 solutions. The full step-by-step solution to problem: 8P from chapter: B was answered by , our top Physics solution expert on 07/05/17, 04:29AM. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. Since the solution to 8P from B chapter was answered, more than 252 students have viewed the full step-by-step answer. The answer to “Evaluate (Hint: Change variables to convert the integrand to a Gaussian.) Then use the recursion formula to evaluate” is broken down into a number of easy to follow steps, and 18 words. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1.

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Evaluate (Hint: Change variables to convert the integrand

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