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# A logarithmic integral

Prove that

Solution

We begin by making the substitution thus:

Now let us consider the complex function where the principal arguement of lies within the interval as well as the contour below

It is clear that has two poles of order at and . The residue at is equal to whereas the residue at is equal to . Thus

Sending and the contribution of both the large and the small circle is . Hence:

Thus the conclusion follows.