# Notes - Complex Analysis MT23, Paths and curves

> Source: https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/paths-and-curves/ · Updated: 2023-10-11 · Tags: uni, notes

- [Course - Metric Spaces MT23](https://ollybritton.com/notes/uni/part-a/mt23/metric-spaces/)
	- [Notes - Complex Analysis MT23, Complex integration](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/complex-integration/)

### Flashcards
@Define a path in the complex plane.::
A continuous function $\gamma : [a, b] \to \mathbb C$.

@Define what it means 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?::
$$
\mathrm{Im}\, \gamma \subseteq \mathbb C
$$

@Define what it means 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 not exist?::
$$
L(t) = \gamma(t_0) + (t - t_0)\gamma'(t_0)
$$
doesn’t exist when $\gamma'(t_0) = 0$.

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.

@Define 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 $\phi(b) = d$.

@Define what it means 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?)

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

@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?::
$$
\int_\Gamma f(z) \text dz = \sum^k_{i=1} a_i \int_{\gamma_i} f(z) \text d z
$$

@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 $\gamma'(a) \ne 0$. @Define the tangent line $L_a$ on $\gamma$ at $t = a$.::
$$
L_a = \{\gamma(a) + x\gamma'(a) : x \in \mathbb R\}
$$

Suppose $\gamma_1, \gamma_2 : [-1, 1] \to \mathbb C$. @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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