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Due: February 6th, 2020

Math 185 Assignment 2

  1. Products of power series

    1. Let $f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n$ and $g(z) = \sum_{n=0}^\infty d_n(z-z_0)^n$ be power series with radii of convergence $R$ and $S$ respectively. Find a power series expansion for $fg$ centred at $z_0$, and show that its radius of convergence is at least $\min(R, S)$.
    2. Prove that for $w, z \in \C$, we have $e^{w+z} = e^we^z$. Use this and the fact that $e^{iz} = \cos(z) + i\sin(z)$ to deduce the formulae \[\cos(w+z) = \cos(w)\cos(z) - \sin(w)\sin(z) \qquad\text{and}\qquad \sin(w+z) = \cos(w)\sin(z) + \sin(w)\cos(z).\]

  2. Directional derivatives

    Suppose $\Omega \subseteq \C$ is open, $f : \Omega \to \C$, $z\in \Omega$, and $\zeta \in \T \color{red}{ := \set{z \in \C : \abs{z} = 1}}$. The directional derivative of $f$ is given by \[D_\zeta f(z) := \lim_{\substack{t\to0\\t\in\R}} \frac{f(z+t\zeta) - f(z)}t,\] provided the limit exists.

    1. Show that if $f$ is holomorphic at $z$ then $D_\zeta f(z) = \zeta f'(z)$ for each $\zeta \in \T$.

    (Remark: Notice that if $f'(z) \neq 0$, then for $\zeta, \xi \in \T$ we can interpret the ratio $D_\zeta f(z) / D_\xi f(z)$ as the angle between the curves $t\mapsto f(z+t\zeta)$ and $t \mapsto f(z+t\xi)$; but then this is just $\zeta\bar\xi$, i.e., the angle between the lines $t \mapsto z+t\zeta$ and $t\mapsto z+t\xi$. For this reason, we say that $f$ "preserves angles" or "is conformal" at $z$.)

    1. Use (a) to produce another proof that if $f$ is holomorphic then it satisifies the Cauchy-Riemann equations.
    2. Prove the following statement, or provide an explicit counterexample: if there is some $a \in \C$ so that for each $\zeta\in\T$ we have $D_\zeta f(z) = a\zeta$, then $f$ is holomorphic at $z$ and $f'(z) = a$.

  3. The Ratio Test

    Suppose that $(a_n)_n$ is a sequence in $\C \setminus\set0$ such that \[\limni\abs{\frac{a_{n+1}}{a_n}} = L.\] Show that \[\limni \abs{a_n}^{1/n} = L.\] Note that this provides another method of computing the radii of convergence of some power series.

  4. A condition for uniform convergence

    Suppose that $K$ is a compact set and $(f_n)_n$ is a sequence of functions from $K$ to $\C$. Show that $(f_n)_n$ converges uniformly on $K$ if and only if it is uniformly Cauchy, i.e, for every $\epsilon \gt 0$ there is $N \in \N$ so that if $n, m \gt N$ we have $\norm{f_n - f_m}_K \lt \epsilon$.

    (If $(f_n)_n$ is uniformly Cauchy, then for each $z \in K$ the sequence $(f_n(z))_n$ is Cauchy because $\abs{f_n(z) - f_m(z)} \leq \norm{f_n-f_m}_K$. Therefore $(f_n(z))_n$ converges, and so $(f_n)_n$ converges pointwise to a function $f : z \mapsto \limni f_n(z)$. Your goal is to show that this convergence is uniform.)

  5. Behaviour at the radius of convergence is delicate

    Devise a power series $f(z) = \sum_{n=0}^\infty a_nz^n$ with radius of convergence 1 which converges uniformly on $\overline{B_1(0)} = \set{z \in \C : \abs{z} \leq 1}$, but the power series describing $f''(z)$ converges at no point of $\T$.

    (If you are stuck, click for a suggestion of some coefficients which will work.) Try $a_n = \frac1{n^2}$ for $n \gt 0$. Of course, you still need to prove that these have the desired property.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.