Homework 4 collaboration area
This is the place.
Exercise 1
For what values of $ z $ is the series
$ \sum_{n=0}^{\infty}\left(\frac{z}{1+z}\right)^{n} $
convergent? Same question for
$ \sum_{n=0}^{\infty}\frac{z^{n}}{1+z^{2n}}. $
Discussion
-The first series yields readily to an application of Hadamard's formula and a subsequent analysis of those values $ z $ for which the limsup is equal to $ 1 $. The second series seems a little more difficult. Neither Hadamard nor the ratio test seem to be very easy to work out. Can we recognize these terms as the derivative of another series? We know that the derivative of a power series will have the same radius of convergence as the original series. Should we try to decompose the series somehow? For example, we can write
$ \sum_{n=0}^{\infty}\frac{z^{n}}{1+z^{2n}}=\sum_{n=0}^{\infty}\left(\frac{\frac{1}{2}z^{n}}{z^{n}-i}+\frac{\frac{1}{2}z^{n}}{z^{n}+i}\right). $
The trouble with this idea is that we know that if the two series in this decompostion converge, then their sum converges. The converse isn't true though...
Exercise 2
If $ f $ is analytic on the unit disc and
$ \left| f(z)\right|\leq \frac{1}{1-\left| z\right|}, $
find the best estimate of
$ \left| f^{(n)}(0)\right| $
thet the Cauchy Estimates will yield.
Discussion
-The exercise seems relatively straightforward: Express the bound given by the Cauchy Estimates as a function of one real variable and then use standard calculus optimization techniques to find the minimum. Anyone have a different idea? Any slick proofs?
Exercise 3
Show that the successive derivatives of an analytic function at a point $ a $ can never satisfy
$ \left|f^{(n)}(a)\right|>n!n^{n}. $
Formulate a sharper theorem of the same kind.
From Steve Bell 13:29, 18 February 2014 (UTC): Just to be clear, the function f(z) in this problem is assumed to be analytic on an open set containing the origin.
Discussion
-It seems the key here is to note that an analytic function can be locally expressed by a power series which converges on a disc of positive radius. The coefficients of this power series relate to the quantities in question. How precise does this "sharper theorem" need to be? There seems to be an obvious choice given the proof of the result asked for.
Exercise 4
Prove that
$ \cos(\theta+\psi)=\cos\theta\cos\psi-\sin\theta\sin\psi $
without mentioning trigonometry or angles.
Discussion
Exercise 5
Suppose that $ f $ is analytic on a disk $ D_{\epsilon}(0) $ and satisfies the differential equation
$ f^{\prime\prime}=f. $
Prove that $ f $ is given by
$ A\cosh z+B\sinh z, $
where $ A $ and $ B $ are constants.
Discussion
Can any one tell me what is the formal definition of unbounded complex valued function?
-I would say that an unbounded complex valued function is a function
$ f:X\rightarrow \mathbb{C} $
from a set $ X $ into the complex numbers with the property that
$ \left|f\right| $
is an unbounded function. This says that for any real number $ M $, there is an element
$ x\in X $
such that
$ \left|f(x)\right|>M. $
Intuitively, this means that the image of the function cannot be contained in a big ball. I hope this helps.
