===== Complex line integral =====
==== Function ====
| @#DDDDDD: range       | @#DDDDDD: $\mathcal{L}$ ... continuously differentiable finite lines |
| @#FFBB00: definiendum | @#FFBB00: $\int: \mathcal{L}\to(\mathbb C\to \mathbb C)\to \mathbb K$ |
| @#DDDDDD: range       | @#DDDDDD: $L\in \mathcal{L}$ |
| @#DDDDDD: range       | @#DDDDDD: $\gamma: [a,b]\to L$ ... parametrization |
| @#FFBB00: definiendum | @#FFBB00: $\int_L\ f(z)\,\mathrm dz:=\int_L\ f\left(\gamma(t)\right)\cdot \gamma'(t)\, \mathrm dt$ |

>todo: [[Continuously differentiable finite lines]]

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=== 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 ==
^ $\int_L f'(z)\,\mathrm dz=f(b)-f(a)$ ^

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=== Requirements ===
[[Function integral]], [[Integral over a subset]]