Home » Posts tagged 'Complex Analysis'
Tag Archives: Complex Analysis
Consider the points and . Let be a point of the plane such that . Set to be the angle that is defined by and . ( the one that is less than .) Prove that the function is harmonic.
Solution [by Demetres Skouteris]
Let be holomorphic on the open unit disk and suppose that the integral converges. If the Taylor expansion of is of the form then prove that the sum
and thus the series converges.
Note: The set of functions satisfying this is a Hilbert space of functions, but it is not the same as the Hardy space .
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 such that . Prove that: