Argument Principle

Argument Principle (Local Ver)

Let $Δ \subseteq C$ be an open ball and $a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m} \in Δ$ . If $f$ is a meromorphic on $Δ$ such that $a_{j}$ is a zero of order $k_{j} \in N^{+}$ , for each $j \in {1, \dots, n}$ and $b_{l}$ is a pole of $f$ of order $h_{l}$ for $l \in {1, \dots, m}$ , then

\frac{1}{2 π i} \oint_{γ} \frac{f^{'} (z)}{f (z)} d z = \sum_{j = 1}^{n} k_{j} \cdot n (γ, a_{i}) - \sum_{l = 1}^{m} h_{l} \cdot n (γ, b_{l})

for all $γ \subseteq Δ ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that is piecewise smooth closed curve

Cor: Let $Δ \subseteq C$ be an open ball and $a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m} \in Δ$ . If $f$ is a meromorphic on $Δ$ such that $a_{j}$ is a zero of order $k_{j} \in N^{+}$ , for each $j \in {1, \dots, n}$ and $b_{l}$ is a pole of $f$ of order $h_{l}$ for $l \in {1, \dots, m}$ . If $γ \subseteq Δ ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that is piecewise smooth closed curve and $\tilde{γ} = f \circ γ$ , then

n (\tilde{γ}; 0) = \frac{1}{2 π i} \oint_{γ} \frac{f^{'} (z)}{f (z)} d z = \sum_{j = 1}^{n} k_{j} \cdot n (γ, a_{i}) - \sum_{l = 1}^{m} h_{l} \cdot n (γ, b_{l})

n (\tilde{γ}; 0) = (# of zeros - # of poles) of f trapped by γ

Th (Ahlfors): Let $f : Ω \to C$ analytic, $z_{0} \in Δ \subseteq Ω$ , $Δ$ an open ball, and $w_{0} = f (z_{0})$ . If $z_{0}$ is a zero of order $k$ of the function $g (z) = f (z) - w_{0}$ for all $z \in Δ$ , then there exist $ε, δ > 0$ such that $w \in B_{ε} (w_{0}) ∖ {w_{0}}$ the identity $f (z) = w$ has exactly $k$ solutions in $B_{δ} (z_{0}) ∖ {z_{0}}$ . Additionally, we can find $δ > 0$ such that all the soltions of the identity $f (z) = w$ are different from each other.

Cor: Let $f : Ω \to C$ analytic and $U \subseteq Ω$ be an open subset. If $f$ is nonconstant, then $f [U] \subseteq C$ is an open set.

Cor: Let $f : Ω \to C$ analytic. If $f$ is injective in the region $Ω$ , $f^{'} (z) \neq 0$ for all $z \in Ω$

Prop: Let $f : Ω \to C$ analytic. If $f$ injective in the region $Ω$ , then the inverse function $f^{- 1} : f [Ω] \to Ω$ is analytic in $f [Ω]$ (is also a region) and additionally $(f^{- 1})^{'} (w) \neq 0$ for all $w \in Ω$

Prop: Let $f : Ω \to C$ analytic and injective in $Ω$ . If $z_{0} \in Ω$ and $R > 0$ such that $B_{R} (z_{0}) \subseteq Ω$ and $γ_{r} (t) = r e^{i t} + z_{0}$ , with $t \in [0, 2 π]$ and $0 < r < R$ , then

f^{- 1} (w) = \frac{1}{2 π i} \oint_{γ_{r}} ζ \frac{f^{'} (ζ)}{f (ζ) - w} d ζ

Inverse Function Theorem for Analytic Functions

Let $f : Ω \to C$ analytic. If $z_{0} \in Ω$ is such that $f^{'} (z_{0}) \neq 0$ , then there’s $r > 0$ such that $f$ is injective on $B_{r} (z_{0})$ , $f [B_{r} (z_{0})]$ is open, $f^{- 1}$ is analytic on $f [B_{r} (z_{0})]$ and

(f^{- 1})^{'} (w) = \frac{1}{f^{'} (f^{- 1} (w))} \neq 0

for all $w \in f^{- 1} [B_{r} (w_{0})]$

Rouché-Gliksberg Theorem

Let $f$ and $g$ be meromorphic functions on the disk $Δ \subseteq C$ and $γ \subseteq Δ$ be a piecewise smooth closed curve such that $n (γ; z) = 0$ or $n (γ; z) = 1$ for all $z \in Δ ∖ γ$ . Let $Z_{f}$ be the number of zeros of $f$ enclosed by $γ$ counted with multiplicities, $P_{f}$ the number of poles of $f$ enclosed by $γ$ counted with multiplicities, and $Z_{g}$ and $P_{g}$ be the corresponding ones for the function $g$ . If $f$ and $g$ are such that

| f (z) + g (z) | < | f (z) | + | g (z) |

for all $z \in γ$ , then

Z_{f} - P_{f} = Z_{g} - P_{g}

Rouché Theorem

Let $f$ and $g$ be analytic functions on the disk $Δ \subseteq C$ and $γ \subseteq Δ$ be a piecewise smooth closed curve such that $n (γ; z) = 0$ or $n (γ; z) = 1$ for all $z \in Δ ∖ γ$ . Let $Z_{f}$ be the number of zeros of $f$ enclosed by $γ$ counted with multiplicities, $P_{f}$ the number of poles of $f$ enclosed by $γ$ counted with multiplicities, and $Z_{g}$ and $P_{g}$ be the corresponding ones for the function $g$ . If $f$ and $g$ are such that

| f (z) - g (z) | < | g (z) |

for all $z \in γ$ , then

Z_{f} = Z_{g}

Hurwitz's Theorem

Let $f_{n}, f : Δ \to C$ analytics on the disk $Δ$ and $γ \subseteq Δ$ be a piecewise smooth closed curve such that $n (γ; z) = 0$ or $n (γ; z) = 1$ for all $z \in Δ ∖ γ$ . If the the sequence $f_{n}$ converges uniformly to $f$ over $γ$ and $f (z) \neq 0$ for all $z \in γ$ , then there exists $N \in N$ such that for all $n \geq N$ the functions $f_{n}$ and $f$ have the same amount of zeros enclosed by $γ$

Prop: Let $f_{n} : Ω \to C$ be a sequence of analytic functions such that it converges uniformly to $f$ over any compact set $K \subseteq Ω$ . If $f_{n} (z) \neq 0$ for all $n \in N$ and for all $z \in Ω$ , then $f$ is the constant $0$ or $f (z) \neq 0$ for all $z \in Ω$

Argument Principle

Let $Ω \subseteq C$ be an region and $a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m} \in Ω$ . If $f$ is a meromorphic on $Ω$ such that $a_{j}$ is a zero of order $k_{j} \in N^{+}$ , for each $j \in {1, \dots, n}$ and $b_{l}$ is a pole of $f$ of order $h_{l}$ for $l \in {1, \dots, m}$ , then

\frac{1}{2 π i} \oint_{γ} \frac{f^{'} (z)}{f (z)} d z = \sum_{j = 1}^{n} k_{j} \cdot n (γ, a_{i}) - \sum_{l = 1}^{m} h_{l} \cdot n (γ, b_{l})

for any cycle $γ \subseteq Ω ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that is $γ \sim 0 (\mod Ω)$ . This can be done using the Residue Theorem

Cor: Let $Ω \subseteq C$ be an open ball and $a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m} \in Ω$ . If $f$ is a meromorphic on $Ω$ such that $a_{j}$ is a zero of order $k_{j} \in N^{+}$ , for each $j \in {1, \dots, n}$ and $b_{l}$ is a pole of $f$ of order $h_{l}$ for $l \in {1, \dots, m}$ . If $γ \subseteq Ω ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that cycle that $γ \sim 0 (\mod Ω)$ and $\tilde{γ} = f \circ γ$ , then

n (\tilde{γ}; 0) = \frac{1}{2 π i} \oint_{γ} \frac{f^{'} (z)}{f (z)} d z = \sum_{j = 1}^{n} k_{j} \cdot n (γ, a_{i}) - \sum_{l = 1}^{m} h_{l} \cdot n (γ, b_{l})

Cor: Let $Ω \subseteq C$ be an open ball and $a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m} \in Ω$ . If $f$ is a meromorphic on $Ω$ such that $a_{j}$ is a zero of order $k_{j} \in N^{+}$ , for each $j \in {1, \dots, n}$ and $b_{l}$ is a pole of $f$ of order $h_{l}$ for $l \in {1, \dots, m}$ . If $γ \subseteq Ω ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that is cycle such that $γ \sim 0 (\mod Ω)$ and $n (γ; z)$ can only take the values of $0$ or $1$ and $\tilde{γ} = f \circ γ$ , then

n (\tilde{γ}; 0) = (# of zeros - # of poles) of f trapped by γ

We can extend this to get that:

\frac{1}{2 π i} \oint_{γ} g (z) \frac{f^{'} (z)}{f (z)} d z = \sum_{j = 1}^{n} k_{j} \cdot n (γ, a_{j}) \cdot g (a_{j}) - \sum_{l = 1}^{m} h_{l} \cdot n (γ, b_{l}) \cdot g (b_{l})

for any cycle $γ \subseteq Ω ∖ {a_{1}, \dots, a_{n}, b_{1}, \dots, b_{m}}$ that is $γ \sim 0 (\mod Ω)$