Prove that there exists no rational function such that
The limit of at infinity is finite and call that . Hence:
and of course this contradicts the fact that
where is the Euler – Mascheroni constant .
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
Now one of the ‘s definition is
Let . Prove that
which is a simple application of residues.
Thus making use of parity we have that
Prove that there does not exist a sequence of continuous functions such that converges pointwise, to the function , where is the characteristic polynomial of the rationals in .
It is known that pointwise limits of continuous functions have a meagre set of points of continuity. However, this function is discontinuous everywhere and thus we cannot expect a sequence of continuous functions to converge pointwise to it.