Find the terms through third order and the radius of convergence of the power series for each of the following functions, centered at \(z_0\text{.}\) (Do not find the general form for the coefficients.)
Consider \(f: \R \to \R\) given by \(f(x) := \frac 1 { x^2
+ 1 }\text{,}\) the real version of our function in Example 8.1.11, to show that Corollary 8.1.10 has no analogue in \(\R\text{.}\) 1
Incidentally, the same example shows, once more, that Liouville’s theorem (Corollary 5.3.4) has no analogue in \(\R\text{.}\)
Prove the following generalization of Theorem 8.1.1: Suppose \((f_n)\) is a sequence of functions that are holomorphic in a region \(G\text{,}\) and \((f_n)\) converges uniformly to \(f\) on \(G\text{.}\) Then \(f\) is holomorphic in \(G\text{.}\) (This result is called the Weierstraß convergence theorem.)
Use the previous exercise and Corollary 8.1.13 to prove the following: Suppose \((f_n)\) is a sequence of functions that are holomorphic in a region \(G\) and that \((f_n)\) converges uniformly to \(f\) on \(G\text{.}\) Then for any \(k \in \N\text{,}\) the sequence of \(k\)th derivatives \(\left(f_n^{(k)}\right)\) converges (pointwise) to \(f^{(k)}\text{.}\)
Suppose \(G\) is a region and \(f: G \to \C\) is holomorphic. Prove that the sets
\begin{align*}
X \amp \ = \ \left\{ a \in G : \text{ there exists } r
\text{ such that } f(z) = 0 \text{ for all } z \in D[a,r]
\right\}\\
Y \amp \ = \ \left\{ a \in G : \text{ there exists } r
\text{ such that } f(z) \ne 0 \text{ for all } z \in
D[a,r] \setminus \{ a \} \right\}
\end{align*}
Prove the Minimum-Modulus Theorem (Corollary 8.2.6): Suppose \(f\) is holomorphic and nonconstant in a region \(G\text{.}\) Then \(|f|\) does not attain a weak relative minimum at a point \(a\) in \(G\) unless \(f(a)=0\text{.}\)
If \(G\) is simply connected then apply Theorem 8.2.4 to \(\exp(u(z)+iv(z)))\text{,}\) where \(v\) is a harmonic conjugate of \(u\text{.}\) Conclude that \(u\) is constant on \(G\text{.}\)
If \(G\) is not simply connected, then the above argument applies to \(u\) on any disk \(D[a,R] \subset G\text{.}\) Conclude that the partials \(u_x\) and \(u_y\) are zero on \(G\text{,}\) and adapt the argument of Theorem 2.4.2 to show that \(u\) is constant.
Use Proposition 5.3.1 to show that a polynomial does not achieve its minimum modulus on a large circle; then use the Minimum-Modulus Theorem to deduce that the polynomial has a zero.
Give another proof of (a variant of) the Maximum-Modulus Theorem 8.2.4 via Corollary 8.1.12, as follows: Suppose \(f\) is holomorphic in a region containing \(\overline D[a,r]\text{,}\) and \(|f(z)| \le M\) for \(z \in C[a,r]\text{.}\) Given a point \(z_0 \in D[a,r]\text{,}\) show (e.g., by Corollary 8.1.12) that there is a constant \(c \in \C\) such that
Suppose that \(f\) is holomorphic at \(z_0\text{,}\)\(f(z_0)=0\text{,}\) and \(f'(z_0)\ne0\text{.}\) Show that \(f\) has a zero of multiplicity \(1\) at \(z_0\text{.}\)
for some \(R_1\text{,}\) and that the convergence is uniform in \(\left\{ z \in \C :
\, | z-z_0 | \ge r_1 \right\}\text{,}\) for any fixed \(r_1 > R_1\text{.}\)
Recall that a function \(f: G \to \C\) is even if \(f(-z) = f(z)\) for all \(z \in G\text{,}\) and \(f\) is odd if \(f(-z) = -f(z)\) for all \(z \in G\text{.}\) Prove that, if \(f\) is even (resp., odd), then the Laurent series of \(f\) at 0 has only even (resp., odd) powers.
Suppose \(f\) is holomorphic and not identically zero on an open disk \(D\) centered at \(a\text{,}\) and suppose \(f(a)=0\text{.}\) Use the following outline to show that \(\Re f(z)>0\) for some \(z\) in \(D\text{.}\)
Compute \(\ds \int_\gg \frac{ \exp(z) }{ \sin(z) } \,
\diff{z}\) where \(\gg\) is a closed curve not passing through integer multiples of \(\pi\text{.}\)
