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