# Complex Analysis/Path of Integration
## Smooth paths and path subdivision

The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.
* (WG1) Definition (Smooth path): A path $\gamma :[a,b]\to \mathbb {C}$ is smooth if it is continuously differentiable.
* (UT) Definition (Subdivision): Let $[a,b]$ be an interval, $n\in \mathbb {N}$ and ${a}={u}_{0}<{\ldots }<{u}_{n}={b}$. ${\left({u}_{0},\ldots ,{u}_{n}\right)}\in \mathbb {R} ^{n+1}$ is called a subdivision of ${\left[{a},{b}\right]}$.
* (WG2) Definition (Path subdivision): Let $\gamma :[a,b]\to \mathbb {C}$ be a path in ${U}\subseteq \mathbb {C}$, ${n}\in \mathbb {N}$, ${\left({u}_{0},\ldots ,{u}_{n}\right)}$ a subdivision of $[a,b]$, $\gamma _{k}:{\left[{u}_{{k}-{1}},{u}_{k}\right]}\to \mathbb {C}$ for all ${k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }$ a path in ${U}$. ${\left(\gamma _{1},\ldots ,\gamma _{n}\right)}$ is called a path subdivision of $\gamma$ if $\gamma _{n}{\left({b}\right)}=\gamma {\left({b}\right)}$ and for all ${k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }$ and ${t}\in {\left[{u}_{{k}-{1}},{u}_{k}\right]}$ we have $\gamma _{k}{\left({t}\right)}=\gamma {\left({t}\right)}\wedge \gamma _{k}{\left({u}_{{k}-{1}}\right)}=\gamma _{{k}-{1}}{\left({u}_{k}\right)}$.
* (WG3) Definition (Piecewise smooth path): A path $\gamma :{\left[{a},{b}\right]}\to \mathbb {C}$ is piecewise smooth if there exists a path subdivision ${\left(\gamma _{1},\ldots \gamma _{n}\right)}$ of $\gamma$ consisting of smooth paths $\gamma _{k}$ for all ${k}\in {\left\lbrace {1},\ldots ,{n}\right\rbrace }$.

## Integration path

* (WG4) Definition (Path integral): Let $f:U\to \mathbb {C}$ be a continuous function and $\gamma :[a,b]\to U$ a smooth path, then the path integral is defined as: $\int _{\gamma }f:=\int _{\gamma }f(z)\,dz:=\int _{a}^{b}f(\gamma (t))\cdot \gamma '(t)\,dt$. If $\gamma$ is only piecewise smooth with respect to a path subdivision $(\gamma _{1},\ldots ,\gamma _{n})$, then we define $\int _{\gamma }f(z)\,dz:=\sum _{k=1}^{n}\int _{\gamma _{k}}f(z)\,dz$.
* Definition (Integration path): An integration path is a piecewise smooth (piecewise continuously differentiable) path.

## Example

The following path is piecewise continuously differentiable (smooth) and for the vertices $z_{1},z_{2},z_{3}\in {\text{Spur}}(\gamma )$ the closed triangle path $\gamma :[0,3]\to \mathbb {C}$ is not differentiable. The triangle path is defined on the interval $[0,3]$ as follows:
    $\gamma (t):=\left\langle z_{1},z_{2},z_{3}\right\rangle (t):={\begin{cases}(1-t)\cdot z_{1}+t\cdot z_{2}&{\text{for }}t\in [0,1]\\(2-t)\cdot z_{2}+(t-1)\cdot z_{3}&{\text{for }}t\in (1,2]\\(3-t)\cdot z_{3}+(t-2)\cdot z_{1}&{\text{for }}t\in (2,3]\\\end{cases}}$

### Paths from convex combinations

The piecewise continuously differentiable path is formed from convex combination.The sub-paths
* $\gamma _{1}:=\left\langle z_{1},z_{2}\right\rangle$ with $\gamma _{1}:[0,1]\to \mathbb {C} ,\ (1-t)\cdot z_{1}+t\cdot z_{2}$
* $\gamma _{2}:=\left\langle z_{2},z_{3}\right\rangle$ with $\gamma _{2}:[1,2]\to \mathbb {C} ,\ (2-t)\cdot z_{2}+(t-1)\cdot z_{3}$
* $\gamma _{3}:=\left\langle z_{3},z_{1}\right\rangle$ with $\gamma _{3}:[2,3]\to \mathbb {C} ,\ (3-t)\cdot z_{3}+(t-2)\cdot z_{1}$
are continuously differentiable.