# Wednesday January 22nd

## Parameterized Curves

> Note: multiple complex variables, see Hormander or Steven Krantz

Recall from last time that if $$f(z) = \sum_{n=0}^\infty a_n z^n$$ with $z_0 \neq 0$ has radius of convergence $$R = (\limsup \abs{a_n}^{1/n})\inv > 0$$ then
$f'$ exists and is obtained by differentiating term-by-term.

We know that $f$ analytic $\implies$ $f$ holomorphic (and smooth), and we want to show the converse.
For this, we need integration.

Definition (Parameterized Curves)
: 	A *parameterized curve* is a function $z(t)$ which maps a closed interval $[a, b] \subset \RR$ to $\CC$.

Definition (Smooth Curves)
: 	The curve is said to be *smooth* iff $z'$ exists and is continuous on $[a,b]$, and $z'(t) \neq 0$ for any $t$.
	At the boundary $\theset{a, b}$, we define the derivative by taking one-sided limits.

Definition (Piecewise Smooth Curves)
: 	A curve is said to be *piecewise smooth* iff $z(t)$ is continuous on $[a, b]$ and there are $a < a_1 < \cdots < a_n = b$ with $z$ smooth on each $[a_k, a_{k+1}]$.

Note that such a curve may fail to have tangent lines at $a_i$.

Definition (Equivalent Parameterizations)
: 	Two parameterizations $z: [a,b] \to \CC, \tilde z: [c, d] \to \CC$ are *equivalent* iff there exists a $C^1$ bijection $s: [c, d] \to [a, b]$ where $s \mapsto t(s)$ such that $s'>0$ and $\tilde z(s) = z(s(t))$.

Note that $s' > 0$ preserves orientation and $s'<0$ reverses orientation.

Definition (Orientations of Curves)
: A curve in *reverse orientation* is defined by
	\begin{align*}
	\gamma: [a, b] \to \CC \implies 
	\gamma^-: [a,b] &\to \CC \\ t &\mapsto \gamma(a+b-t)
	.\end{align*}

Definition (Closed Curves)
: 	A curve is *closed* iff $z(a) = z(b)$, and is simple iff $z(t) \neq z_{t_1}$ for $t\neq t_1$.

Definition (Positively Oriented Curves)
: 	For $C_r(z_0) \definedas \theset{z\suchthat \abs{z-z_0} = r}$, the *positive orientation* is given by $z(t) = z_0 + re^{2\pi i t}$ for $t\in [0, 1]$.

## Definition of the Integral

Definition (The Complex Integral)
: 	The *integral* of $f$ over $\gamma$ is defined as

	\begin{align*}
	\int_\gamma f ~dz = \int_a^b f(z(t)) z'(t)~dt
	.\end{align*}

Note: this doesn't depend on parameterization, since if $t = t(s)$, then a change of variables yields
\begin{align*}
\int_\gamma f ~dz - \int_c^d f(z(t(s)))~z'(t(s))~t'(s) ~ds = \int_c^d f(\tilde z(s)) ~\tilde z'(s) ~ds
.\end{align*}

Definition (Length of a Curve)
: 	The *length* of $\gamma$ is defined as $\abs \gamma = \int \abs{z'(t)} ~dt$.

Proposition
: 	1. We can extend this definition to piecewise smooth curves by 
	$$
	\int_\gamma f~dz = \sum \int_{a_k}^{a_{k+1}} f ~dz
	$$

	2. This integral is linear and $\int_\gamma f = -\int_{\gamma^-} f$.

	3. We have an inequality

	\begin{align*}
	\abs{\int_\gamma f} \leq \max_{a\leq t \leq b} \abs{f(z(t))} \abs\gamma
	.\end{align*}

Definition (Primitive of a Function)
: 	A function $F$ is a *primitive* for $f$ on $\Omega$ iff $F$ is holomorphic on $\Omega$ and $F'(z) = f(z)$ on $\Omega$.

Recall that in $\RR$, we have $$F(x) =\int_a^x f(t)~dt$$ as an antiderivative with $F'(x) = f(x)$, and $\int f = F(b) - F(a)$.

Theorem (Evaluating Integrals with Primitives)
: 	If $f$ is continuous, has a primitive $F$ in $\Omega$, and $\gamma$ is a curve beginning at $w_0$ and ending at $w_1$, then $\int_\gamma f = F(w_1) - F(w_0)$.

Proof
: 	Use definitions, write $z(t)$ where $z(a) = w_1, z(b) = w_2$.
	Then

	\begin{align*}
	\int_\gamma f &= \int_a^b f(z(t)) z'(t) ~ dt \\
	&= \int_a^b F'(z(t)) z'(t) ~dt \\
	&= \int_a^b F_t ~dt \\
	&= F(z(b)) - F(z(a)) \quad\text{by FTC}\\
	&= F(w_1) - F(w_2)
	.\end{align*}

	Note that if $\gamma$ is piecewise smooth, the sum of the integrals telescopes to yield the same conclusion.

Corollary (Functions with Primitives Integrate to Zero Along Loops)
: 	If $f$ is continuous and $\gamma$ is a closed curve in $\Omega$, and $f$ has a primitive in $\Omega$, then $$\oint f = 0.$$