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Due: February 27th, 2020
Math 185 Assignment 5
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Limits of holomorphic functions
Use Morera's theorem to prove the following.
Suppose that $\Omega \subseteq \C$ is open, and $(f_n)_n$ is a sequence of holomorphic functions which converge uniformly to $f$ on compact subsets of $\Omega$.
Then $f$ is holomorphic.
(You may assume without proof the fact that if $T \subset \Omega$ is a triangle, then there is a convex open set $U$ with $T \subseteq U \subseteq \Omega$.
This follows from essentially the same argument as problem 1 on assignment 4.)
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Bounds on functions
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Suppose that $f : \C \to \C$ is entire with the property that for some $C \gt 0$, for all $z \in \C$,
\[\abs{f(z)} \leq Ce^{\Re(z)}.\]
Prove that there is a constant $c \in \C$ so that $f(z) = ce^z$.
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Suppose that $f : \C \to \C$ is entire.
Show that $f$ is a polynomial if and only if there exist a constant $C \gt 0$ and some $n \in \N$ such that for all $z \in \C$
\[\abs{f(z)} \leq C(1 + \abs{z})^n.\]
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Holomorphic extensions of real functions
Suppose $f : \R \to \R$, and $\Omega \subseteq \C$ is a domain with $\R \subset \Omega$.
Prove that $f$ has at most one holomorphic extension to $\Omega$.
(That is, show that if there is a holomorphic function $g : \Omega \to \C$ so that $g|_\R = f$, then $g$ is unique.)
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Removable singularities and a characterization of sine
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Suppose that $\Omega \subseteq \C$ is open, $f, g : \Omega \to \C$ are holomorphic, and $z_0 \in \Omega$ is a zero of order $n$ for $f$ and of order $m$ for $g$, for some $1 \leq m \leq n$.
Show that $z_0$ is a removable singularity for $\frac fg$.
(You may assume without proof that $\frac fg$ is defined on an appropriate punctured neighbourhood of $z_0$, but you might wish to reflect for a moment on why that is the case.)
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Suppose that $f : \C \to \C$ is holomorphic, with $f(n) = 0$ for each $n \in \Z$.
Show that each singularity of $z \mapsto \frac{f(z)}{\sin(\pi z)}$ is removable.
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Suppose that $f : \C \to \C$ is holomorphic and satisfies $f(z+1) = -f(z)$, $f(0) = 0$, and for some $C \gt 0$, $\abs{f(z)} \leq C\exp(\pi\abs{\Im(z)})$.
Show that $f(z) = c\sin($$\pi$$z)$ for some $c \in \C$.
(Correction: there was a missing $\pi$ in the statement of the problem, and $f$ was meant to be holomorphic.)
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.
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Fives
Suppose that $\Omega \subseteq \C$ is open and $f : \Omega \to \C$ is holomorphic.
Let us say that $z_0 \in \Omega$ is a five of $f$ of order $N$ if $f^{(N)}(z_0) \neq 5$ but for all $n \lt N$,
\[f^{(n)}(z_0) = 5.\]
Suppose that $f : \Omega \to \C$ has a five of order $N$ at $z_0 \in \Omega$.
Prove that there is a holomorphic function $h : \Omega \to \C$ with $h^{(N)}(z_0) \neq 5$ so that
\[f(z) = \sum_{n=0}^{N-1} \frac{5z^n}{n!} + \frac{(z-z_0)^N}{N!} h(z).\]
Moreover, show that if $z_0$ is a five of infinite order (i.e., if $f^{(n)}(z_0) = 5$ for all $n \in \N_0$), then
\[f(z) = 5\exp(z-z_0).\]
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A bound on the derivative
Let us write $\D = B_1(0) = \set{z \in \C\mid \abs{z} \lt 1}$.
Suppose that $f : \D \to \C$ is holomorphic.
Show that
\[2\abs{f'(0)} \leq \sup\set{\abs{f(z) - f(w)} \mid z, w \in \D}.\]
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More bounds
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Suppose that $f : \C \to \C$ is such that $\Re(f(z)) \leq M$ for all $z\in \C$ and some constant $M \in \R$.
Prove that $f$ is constant.