31.1 Complex Differentiability

Lemma 31.1.1.label Let $E$ be a separated locally convex space over $\complex$, $U \subset \complex$, and $f: U \to E$, then the following are equivalent:

(1)
$f \in C^{1}(U; E)$.
(2)
Under the identification of $C = \real^{2}$, $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\in C(U; E)$ and
\[\frac{\partial f}{\partial x}= i\frac{\partial f}{\partial y}\]

Proof. (1) $\Rightarrow$ (2): Let $x_{0} \in U$, then

\[\frac{\partial f}{\partial x}= \lim_{\substack{h \to 0 \\ h \in \real}}\frac{f(x_{0} + h) - f(x_{0})}{h}= \lim_{h \to 0}\lim_{\substack{h \to 0 \\ h \in \real}}\frac{f(x_{0} + ih) - f(x_{0})}{ih}= \frac{1}{i}\frac{\partial f}{\partial y}\]

(2) $\Rightarrow$ (1): Let $x_{0} \in U$ and

\[L: \complex \to E \quad a + bi \mapsto a \frac{\partial f}{\partial x}(x_{0}) + b \frac{\partial f}{\partial y}(x_{0})\]

by assumption and Proposition 11.5.1, $L \in L(\complex; E)$. By Proposition 30.6.2, $f \in C^{1}(U \subset \real^{2}; E)$, where for any $(a, b) \in \real^{2}$,

\[Df(x_{0})(a, b) = a \frac{\partial f}{\partial x}(x_{0}) + b \frac{\partial f}{\partial y}(x_{0})\]

so by definition of differentiability, $f$ is complex-differentiable at $x_{0}$ with derivative $L$.$\square$

Theorem 31.1.2 (Cauchy).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $f \in C^{1}(U; E)$, and $\gamma, \mu \in C([a, b]; U)$ be closed rectifiable paths. If $\gamma$ and $\mu$ are homotopic, then

\[\int_{\gamma} f = \int_{\mu} f\]

Proof of smooth case. Let $\Gamma \in C^{\infty}([0, 1] \times [a, b]; U)$ be a smooth homotopy of loops from $\gamma$ to $\mu$, and

\[F: [0, 1] \to E \quad t \mapsto \int_{\Gamma (t, \cdot)}f = \int_{a}^{b} (f \circ \Gamma)(t, s) \Gamma(t, ds)\]

then for any $t \in [0, 1]$, by the change of variables formula,

\begin{align*}F(t)&= \int_{a}^{b} (f \circ \Gamma)(t, s) \Gamma(t, ds) \\&= \int_{a}^{b} (f \circ \Gamma)(t, s) \frac{\partial \Gamma}{\partial s}(t, s) ds\end{align*}

Now, by Proposition 30.9.2,

\[\frac{dF}{dt}(t) = \int_{a}^{b} \frac{\partial}{\partial t}\braks{(f \circ \Gamma)(t, s) \frac{\partial \Gamma}{\partial s}(t, s)}ds\]

Under the identification that $\complex = \real^{2}$, by the power rule and the chain rule,

\[\frac{\partial }{\partial t}\braks{(f \circ \Gamma) \frac{\partial \Gamma}{\partial s}}= (Df \circ \Gamma)\paren{\frac{\partial\Gamma}{\partial t}}\frac{\partial \Gamma}{\partial s}+ (f \circ \Gamma) \frac{\partial^{2}\Gamma}{\partial t \partial s}\]

Now, since $f \in C^{1}(U; E)$ satisfies the Cauchy-Riemann equations,

\begin{align*}(Df \circ \Gamma)\paren{\frac{\partial\Gamma}{\partial t}}\frac{\partial \Gamma}{\partial s}&= (Df \circ \Gamma)\frac{\partial\Gamma}{\partial t}\frac{\partial \Gamma}{\partial s}= (Df \circ \Gamma)\paren{\frac{\partial \Gamma}{\partial s}}\frac{\partial\Gamma}{\partial t}\end{align*}

\begin{align*}\frac{\partial }{\partial t}\braks{(f \circ \Gamma) \frac{\partial \Gamma}{\partial s}}&= (Df \circ \Gamma)\paren{\frac{\partial\Gamma}{\partial s}}\frac{\partial \Gamma}{\partial t}+ (f \circ \Gamma) \frac{\partial^{2}\Gamma}{\partial s \partial t}\\&= \frac{\partial }{\partial s}\braks{(f \circ \Gamma) \frac{\partial \Gamma}{\partial t}}\end{align*}

Hence by the Fundamental Theorem of Calculus,

\begin{align*}\frac{dF}{dt}(t)&= \int_{a}^{b} \frac{\partial}{\partial s}\braks{(f \circ \Gamma)(t, s) \frac{\partial \Gamma}{\partial t}(t, s)}ds \\&= (f \circ \Gamma)(t, b)\frac{\partial \Gamma}{\partial t}(t, b) - (f \circ \Gamma)(t, a)\frac{\partial \Gamma}{\partial t}(t, a)\end{align*}

Since $\Gamma(t, a) = \Gamma(t, b)$ for all $t \in [0, 1]$, the above expression evaluates to $0$, so

\[\int_{\gamma} f = F(0) = F(1) = \int_{\mu} f\]

by Proposition 30.3.6.$\square$

Proof of general case. Let $\Gamma \in C([0, 1] \times [a, b]; \complex)$ be a homotopy of loops from $\gamma$ to $\mu$. By augmenting $\Gamma$ and using Lemma 14.5.5, assume without loss of generality that:

(a)
$\mu$, $\gamma$ are piecewise linear.

Furthermore, by passing through a reparametrisation, assume without loss of generality that:

(b)
For each $t \in [0, \eps)$, $\Gamma(t, \cdot) = \gamma$.
(c)
For each $t \in (1 - \eps, 1]$, $\Gamma(t, \cdot) = \mu$.
(d)
For each $t \in [0, 1]$, $\Gamma$ is constant on $\bracs{t}\times ([a, a + \eps] \cup [b - \eps, b])$.

Extend $\Gamma$ to $[0, 1] \times \real$ by

\[\Gamma_{0}: \real^{2} \to \complex \quad (t, s) \mapsto \begin{cases}\Gamma(t, s) &t \in k(b-a) + [a, b], k \in \integer \\\end{cases}\]

then extend $\Gamma_{0}$ to $\real^{2}$ by

\[\ol \Gamma: \real^{2} \to \complex \quad (t, s) \mapsto \begin{cases}\Gamma(t, s) &t \in [0, 1] \\ \Gamma(1, s) &t \ge 1 \\ \Gamma(0, s) &t \le 0\end{cases}\]

Let $\varphi \in C_{c}^{\infty}(\real^{2}; \real)$ with $\int_{\real^2}\varphi = 1$. For each $\delta \ge 0$, let

\[\Gamma_{\delta}: [0, 1] \times [a, b] \to \complex \quad (t, s) \mapsto \frac{1}{\delta^{2}}\int_{\real^2}\Gamma(y) \varphi\paren{\frac{(t, s) - y}{\delta}}dy\]

Since for each $k \in \integer$ and $(t, s) \in \real^{2}$, $\Gamma(t, s + k(b - a)) = \Gamma(t, s)$, $\Gamma_{\delta}(t, a) = \Gamma_{\delta}(t, b)$ for all $t \in [0, 1]$. Therefore $\Gamma_{\delta}$ is a homotopy of loops. Since $\Gamma$ is continuous, $\Gamma([0, 1] \times [a, b])$ is compact, so $\Gamma_{\delta}$ lies in $U$ for sufficiently small

By assumptions (b) and (c), for sufficiently small $\delta$, there exists $\psi \in C_{c}^{\infty}(\real; \real)$ with $\int_{\real}\psi = 1$ such that

\[\Gamma_{\delta}(0, s) = \frac{1}{\delta}\int_{\real^2}\Gamma(0, y) \psi\paren{\frac{s - y}{\delta}}dy\]

and

\[\Gamma_{\delta}(1, s) = \frac{1}{\delta}\int_{\real^2}\Gamma(1, y) \psi\paren{\frac{s - y}{\delta}}dy\]

By assumption (a), (d), and Lemma 14.5.6,

\[\int_{\gamma} f = \lim_{\delta \downto 0}\int_{\Gamma_\delta(0, \cdot)}f = \lim_{\delta \downto 0}\int_{\Gamma_\delta(1, \cdot)}f = \int_{\mu} f\]

$\square$

Definition 31.1.3.label Let $U \subset \complex$, $z_{0} \in U$, and $r > 0$ such that $\ol{B(z_0, r)}\subset U$, then the path

\[\omega_{z_0, r}: [0, 2\pi] \to U \quad \theta \mapsto a + re^{i\theta}\]

is the standard path of winding number $1$ at $a$ with radius $r$.

Theorem 31.1.4 (Cauchy’s Integral Formula).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $z_{0} \in U$, $r > 0$ such that $\ol{B(z_0, r)}\subset U$, $\gamma \in C([a, b]; \complex)$ be a closed, rectifiable path homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$, and $f \in C^{1}(U; E)$, then

(1)
$\int_{\gamma} f = 0$.
(2)
$f(z) = \frac{1}{2\pi i}\int_{\gamma} \frac{f(z)}{z - z_{0}}dz$.

More over, for any $g \in C(U; E)$ that satisfies (2) for all $z_{0} \in U$, $r > 0$ with $\ol{B(z_0, r)}\subset U$, closed rectifiable curve $\gamma \in C([a, b]; \complex)$ homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$,

(3)
$g \in C^{\infty}(U; E)$, where for each $k \in \natz$,
\[D^{k}g(z_{0}) = \frac{k!}{2\pi i}\int_{\gamma}\frac{g(z)}{(z - z_{0})^{k+1}}dz\]

Proof. By Theorem 31.1.2 and the change of variables formula, for any $g \in C^{1}(U \setminus \bracs{z_0}; E)$,

\[\int_{\gamma} g = \lim_{s \downto 0}\int_{\omega_{z_0, s}}g = \int_{0}^{2\pi}= \lim_{s \downto 0}\frac{s}{2\pi}\int_{0}^{2\pi}g \circ \omega_{z_0, s}(\theta) e^{i\theta}d\theta\]

(1): Since $f \in C(U; E)$, $f$ is bounded on $\ol{B(z_0, r)}$, so for any $s \in (0, r)$,

\[\frac{s}{2\pi}\int_{0}^{2\pi}f \circ \omega_{z_0, s}(\theta) e^{i\theta}d\theta \in s\ol{\text{Conv}}(f(\ol{B(z_0, r)}))\]

As $E$ is locally convex,

\[\int_{\gamma} g = \lim_{s \downto 0}\int_{\omega_{z_0, s}}g = 0\]

(2): Since $f \in C(U; E)$,

\begin{align*}\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z - z_{0}}dz&= \lim_{s \downto 0}\frac{s}{2\pi}\int_{0}^{2\pi}\frac{f \circ \omega_{z_0, s}(\theta)}{\omega_{z_0, s}(\theta) - z_{0}}e^{i\theta}d\theta \\&= \lim_{s \downto 0}\frac{1}{2\pi}\int_{0}^{2\pi}f \circ \omega_{z_0, s}(\theta) d\theta = f(z_{0})\end{align*}

(3): Suppose inductively that (3) holds for $k \in \natz$. For sufficiently small $h \in \complex$,

\[\frac{D^{k}g(z_{0} + h) -D^{k}g(z_{0})}{h}= \frac{k!}{2\pi ih}\int_{\gamma} \frac{g(z)}{(z - z_{0}-h)^{k+1}}- \frac{g(z)}{(z- z_{0})^{k+1}}dz\]

By Proposition 30.9.2,

\[\lim_{h \to 0}\frac{D^{k}g(z_{0} + h) -D^{k}g(z_{0})}{h}= \frac{(k+1)!}{2\pi i}\int_{\gamma} \frac{g(z)}{(z - z_{0})^{k+2}}dz\]

Therefore $g \in C^{k+1}(U; E)$ with

\[D^{k+1}g(z_{0}) = \frac{(k+1)!}{2\pi i}\int_{\gamma} \frac{g(z)}{(z - z_{0})^{k+2}}dz\]

$\square$

Corollary 31.1.5 (Cauchy’s Estimate).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $z_{0} \in U$, $r > 0$ such that $\ol{B(z_0, r)}\subset U$, then for any $k \in \natz$ and continuous seminorm $[\cdot]_{E}: E \to [0, \infty)$,

\[[D^{k}f(z_{0})]_{E} \le \frac{k!}{r^{k}}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E}\]

Proof. By Cauchy’s Integral Formula and Proposition 14.4.1,

\begin{align*}D^{k}f(z_{0})&= \frac{k!}{2\pi i}\int_{\omega_{z_0, r}}\frac{f(z)}{(z - z_{0})^{k+1}}dz \\ [D^{k}f(z_{0})]_{E}&\le \frac{k!}{2\pi i}\int_{0}^{2\pi}\frac{[f(z)]_{E}}{|z - z_{0}|^{k+1}}dz \\&= \frac{k!}{2\pi i}\int_{0}^{2\pi}\frac{[f(z)]_{E}}{r^{k+1}}dz \le \frac{k!}{r^{k}}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E}\end{align*}

$\square$

Definition 31.1.6 (Holomorphic).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, and $f \in C(U; E)$, then the following are equivalent:

(1)
(Complex Differentiability) $f \in C^{1}(U; E)$.
(2)
(Cauchy-Riemann Equations) Under the identification of $C = \real^{2}$, $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\in C(U; E)$ and
\[\frac{\partial f}{\partial x}= i\frac{\partial f}{\partial y}\]
(3)
(Cauchy’s Integral Formula) For each $z_{0} \in U$, $r > 0$ such that $\ol{B(z_0, r)}\subset U$, and closed rectifiable path $\gamma \in C([a, b]; U)$ homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$,
\[f(z_{0}) = \frac{1}{2\pi i}\int_{\gamma} \frac{f(z)}{z - z_{0}}dz\]
(4)
(Analyticity) For each $z_{0} \in U$ and $r > 0$ with $\ol{B(0, r)}\subset U$, there exists $\seq{a_n}\subset E$ such that for each $z \in B(z_{0}, r/2)$,
\[f(z) = \sum_{n = 0}^{\infty} a_{n}(z - z_{0})^{n}\]

where the radius of convergence of the series is at least $r$.
(5)
(Weak Holomorphy) For each $\phi \in E^{*}$, $\phi \circ f$ satisfies the above.

If the above holds, then $f$ is holomorphic/complex analytic on $U$.

Proof. (1) $\Leftrightarrow$ (2): Lemma 31.1.1.

(1) + (2) $\Rightarrow$ (3): See Cauchy’s Integral Formula.

(3) $\Rightarrow$ (4): By Cauchy’s Integral Formula, $f \in C^{\infty}(U; E)$ where for each $k \in \natz$,

\[D^{k}f(z_{0}) = \frac{k!}{2\pi i}\int_{\gamma}\frac{f(z)}{(z - z_{0})^{k+1}}dz\]

Let

\[g(z) = \sum_{k = 0}^{\infty} \frac{1}{k!}D^{k}f(z_{0})(z - z_{0})^{k}\]

then by Cauchy’s Estimate, for any $k \in \natz$ and continuous seminorm $[\cdot]_{E}: E \to [0, \infty)$,

\[[D^{k}f(z_{0})]_{E} \le \frac{k!}{r^{k}}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E} = \frac{Ck!}{r^{k}}\]

Thus $[D^{k}f(z_{0})/k!]_{E} \le C/r^{k}$ for all $k \in \natz$, and the radius of convergence of $g$ is at least $r$.

Let $z \in B(z_{0}, r/2)$, $s = |z - z_{0}|$, and $n \in \natp$, then by Taylor’s Formula and Cauchy’s Estimate,

\begin{align*}\braks{f(z) - \sum_{k = 0}^n \frac{1}{k!} D^kf(z_0)(z - z_0)^n}_{E}&\le s^{n+1}\cdot \sup_{z' \in \ol{B(z_0, s)}}[D^{n+1}f(z')]_{E} \\&\le \frac{Cs^{n+1}}{(r-s)^{n+1}}\end{align*}

which tends to $0$ as $n \to \infty$.

(4) $\Rightarrow$ (1): By Theorem 30.7.3.

(5) $\Rightarrow$ (3): By the equivalence of the prior points, for any $\phi \in E^{*}$, $\phi \circ f$ satisfies (3). By the Hahn-Banach Theorem, $f$ also satisfies (3).$\square$

Remark 31.1.1.label Since weak holomorphy is equivalent to strong holomorphy, most common results of scalar-valued holomorphic functions may be transferred to the vector valued case with little effort. Since this chapter represents my first attempt at learning complex analysis, and I happen to require vector-valued results, most results here are stated and proven in the vector-valued case. However, this is strictly unnecessary if one already knows the scalar-valued results.

Jerry's Digital Garden

31.1 Complex Differentiability

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