Evaluate the integral
Let us now consider the function as well as the contour
The function has three simple poles of which only is included within the contour. The residue at turns out to be . Thus
The contribution of the large circle as is whereas the contribution of the small circle as is . This can be seen by parametrising the small circle ( ). Hence:
Using we get that
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.
Let be a differentiable function such that forall . Does it follow that is necessarily constant?
contradicting what we had assumed in the first place. Hence is constant.
Let such that . Prove that: