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The antiderivative of set equal to zero at x = 0 and

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

Solution for problem 6P 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 6P

The antiderivative ofset equal to zero at x = 0 and multiplied by is called the error function, abbreviated erf x:

(a) Show that erf(±∞) = ±1.

(b) Evaluatein terms of erf x.

(c) Use the result of Problem to find an approximate expression for erf x when x ≫ 1.

Problem:

Evaluate this integral approximately as follows. First, change variables to s = t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2

Step-by-Step Solution:

Step 1 </p>

a) The error function of x is given by

erf x = e-t^2 dt

erf x from 0 to is given by

erf x = e-x^2 dx

Step 2</p>

Expanding the term ex

ex  = 1+ +++..........

e-x  = 1-+++..........

e-2x =1-+++........

By applying this value in erf x

erf x = [ 1-+++........] dx

 erf x =

Where

Then erf x = = 1

Step 3</p>

Similarly

 erf x = e-x^2 dx = )

erf x =-1

So we can conclude erf() =1

Step 4 of 5

Chapter B, Problem 6P is Solved
Step 5 of 5

Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

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