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# The antiderivative of set equal to zero at x = 0 and ISBN: 9780201380279 40

## Solution for problem 6P Chapter B

An Introduction to Thermal Physics | 1st Edition

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Problem 6P

The antiderivative of set equal to zero at x = 0 and multiplied by is called the error function, abbreviated erf x: (a) Show that erf(±∞) = ±1.

(b) Evaluate in 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

Step 5 of 5

##### ISBN: 9780201380279

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