Notes - Complex Analysis MT23, Paths and curves

Created: October 11, 2023

Flashcards

What is a path in the complex plane?

A continuous function $\gamma : [a, b] \to \mathbb C$.

What does it mean for a path $\gamma : [a, b] \to \mathbb C$ to be closed?
\[\gamma(a) = \gamma(b)\]

What does the notation $\gamma^\ast$ mean for a path?
\[\text{Im } \gamma \subseteq \mathbb C\]

What does it mean for a path $\gamma$ to be $C^1$?

It has a continuous derivative.

Suppose we have a $C^1$ path $\gamma$ with derivative $\gamma’$. What is the formula for the tangent to the path, and when does the tangent does not exist?
\[L(t) = \gamma'(t_0) + (t - t_0)\]

doesn’t exist when $\gamma’(t _ 0)$.

Can you give an example of a $C^1$ path without a tangent at some point?
\[\gamma(t) = \begin{cases} t^2 &\text{if } -1 \le t \le 0 \\\\ it^2 &\text{if } 0 \le t \le 1 \end{cases}\]

has no tangent at the origin.

What is a reparameterisation of a $C^1$ path $\gamma : [c, d] \to \mathbb C$?
\[\tilde{\gamma} = \gamma \circ \phi\]

where $\phi : [a, b] \to [c, d]$ a $C^1$ function with $\phi(a) = c$ and $\phi(b) = d$.

What does it mean for two paths $\gamma _ 1 : [a, b] \to \mathbb C$, $\gamma _ 2 : [c, d] \to \mathbb C$ to be equivalent?

If there exists a $C^1$ smooth function $s’(t) > 0$ with $\gamma _ 1 = \gamma _ 2 \circ s$. (can this be rephrased as $\exists$ continuous bijection?)

Can you define the length of a $C^1$ path given by $\gamma : [a, b] \to \mathbb C$?
\[\ell(\gamma) = \int^b_a |\gamma'(t)| \text d t\]

Can you define the integral for a cycle $\gamma = \sum _ {i=1}^{k} a _ i \gamma _ i$, i.e.

\[\int_\Gamma f(z) \text dz\]

where $a _ i \in \mathbb C$ (but usually $\mathbb N$) and each $\gamma _ i$ is a path?
\[\int_\Gamma f(z) \text dz = \sum^k_{i=1} a_i \int_{\gamma_i} f(z) \text d z\]

Can you define the winding number for a cycle $\gamma = \sum _ {i=1}^{k} a _ i \gamma _ i$, i.e.

\[I(\Gamma, z)\]

?
\[\sum^k_{i = 1} a_i I(\gamma_i, z)\]

Suppose $\gamma : [-1, 1] \to \mathbb C$ is a $C^1$ path and that $a \in [-1, 1]$ such that $\gamma’(a) \ne 0$. Can you define the tangent line $L _ a$ on $\gamma$ at $t = a$?
\[L_a = \\{\gamma(a) + x\gamma'(a) : x \in \mathbb R\\}\]

Suppose $\gamma _ 1, \gamma _ 2 : [-1, 1] \to \mathbb C$. Can you define the angle between the two curves at a point $a \in [-1, 1]$?

The difference in argument of the derivatives $\gamma _ 1’(a)$ and $\gamma’ _ 2(a)$ (assuming these are non-zero).

Proofs



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