Contour integral

Contour integrals are the complex analysis equivalent of line integrals. Similar to the real case, we are interested in parameterisation.

$${\int }_{C}f(z)dz={\int }_a^{b}f(z(t))z^{\prime }(t)dt$$ $$dz=z^{\prime }(t)dt$$

For a complex function, we can also define the circulation and flux:

$${\oint }_{C}f\cdot \vec{T}ds={\oint }_{C}udx+vdy$$ $$\oint f\cdot \vec{N}ds={\oint }_{C}udy-vdx$$

Where:

$${\oint }_C\overline{f(z)}dz={\oint }_C\overline{f(z(t))}{z}^{\prime }(t)dt={\oint }_{C}f\cdot \vec{T}ds+i\oint f\cdot \vec{N}ds$$

What this means is that the real part is the circulation and the imaginary part is the flux.

Related concepts

• Analytic function and harmonic function
• Cauchy-Goursat theorem