## Complex line integral

### Function

 range $\mathcal{L}$ … continuously differentiable finite lines definiendum $\int: \mathcal{L}\to(\mathbb C\to \mathbb C)\to \mathbb K$ range $L\in \mathcal{L}$ range $\gamma: [a,b]\to L$ … parametrization definiendum $\int_L\ f(z)\,\mathrm dz:=\int_L\ f\left(\gamma(t)\right)\cdot \gamma'(t)\, \mathrm dt$

#### Theorems

If $f$ is holomorphic and two curves $L_1,L_2$ can be deformed into each other, then

$\int_{L_1} f(z)\,\mathrm dz=\int_{L_2} f(z)\,\mathrm dz$
##### Fundamental theorem of calculus
$\int_L f'(z)\,\mathrm dz=f(b)-f(a)$