Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
complex_line_integral [2014/03/21 11:11] 127.0.0.1 external edit |
complex_line_integral [2015/04/03 12:15] nikolaj |
||
|---|---|---|---|
| Line 2: | Line 2: | ||
| ==== Function ==== | ==== Function ==== | ||
| | @#DDDDDD: range | @#DDDDDD: $\mathcal{L}$ ... continuously differentiable finite lines | | | @#DDDDDD: range | @#DDDDDD: $\mathcal{L}$ ... continuously differentiable finite lines | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $\int: \mathcal{L}\to(\mathbb C\to \mathbb C)\to \mathbb K$ | | | @#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: $L\in \mathcal{L}$ | @#DDDDDD: range | @#DDDDDD: $\gamma: [a,b]\to L$ ... parametrization | | + | | @#DDDDDD: range | @#DDDDDD: $\gamma: [a,b]\to L$ ... parametrization | |
| | @#FFBB00: definiendum | @#FFBB00: $\int_L\ f(z)\,\mathrm dz:=\int_L\ f\left(\gamma(t)\right)\cdot \gamma'(t)\, \mathrm dt$ | | | @#FFBB00: definiendum | @#FFBB00: $\int_L\ f(z)\,\mathrm dz:=\int_L\ f\left(\gamma(t)\right)\cdot \gamma'(t)\, \mathrm dt$ | | ||
| - | ==== Discussion ==== | ||
| >todo: [[Continuously differentiable finite lines]] | >todo: [[Continuously differentiable finite lines]] | ||
| + | |||
| + | ----- | ||
| === Theorems === | === Theorems === | ||
| If $f$ is holomorphic and two curves $L_1,L_2$ can be deformed into each other, then | If $f$ is holomorphic and two curves $L_1,L_2$ can be deformed into each other, then | ||
| Line 16: | Line 15: | ||
| == Fundamental theorem of calculus == | == Fundamental theorem of calculus == | ||
| ^ $\int_L f'(z)\,\mathrm dz=f(b)-f(a)$ ^ | ^ $\int_L f'(z)\,\mathrm dz=f(b)-f(a)$ ^ | ||
| - | ==== Parents ==== | + | |
| + | ----- | ||
| === Requirements === | === Requirements === | ||
| [[Function integral]], [[Integral over a subset]] | [[Function integral]], [[Integral over a subset]] | ||
