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Chapter 11: Q5P (page 548)

Replace x by ix in (9.1) and let t = iuto show that erf(ix) = ierfi(x), where erfi(x) is defined in (9.7).

Short Answer

Expert verified

The expressions are proved below.

Step by step solution

01

Given information.

Error function is given.

02

Definition of an error function.

The error function (also known as the Gauss error function) is a complicated function of a complex variable that is defined as follows:

Error function尾(p,q)=鈭�0鈭�tp-1(1+t)p+qdt.

03

Begin with the definition of an error function.

Write the definition of an error function.

2蟺鈭�0ixe-u2du

Substitute the following in the integral.

s=iuds=idu

Substitute the values in the error function and continue evaluating the integral.

2蟺鈭�0ixe-u2du=2蟺鈭�0xes2(i)du=i2蟺鈭�0xes2du

Hence, proved

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Most popular questions from this chapter

In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro鈥檚 number. Write out ln N! using Stirling鈥檚 formula, compute the approximate value of each term for N = 1023 , and so justify this commonly used approximation.

Prove equation (6.5).

Use equations (3.4) and (11.5) to show that 螕(p)鈻�ppe-p2蟺p(1+112p+...).

Show that for integral n, m,

1/B(n,m)=m(n+m-1n-1)=n(n+m-1m-1)

Hint: See Chapter 1, Section 13C, Problem 13.3.

Use Stirling鈥檚 formula to evaluatelimn鈫�鈭�(2n)!n22n(n!)2.
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