Residue Theorem

Contour integration in the complex plane — ∮ f(z)dz = 2πi Σ Res(f, zₖ)

Contour Radius 2.00

Contour Center Re 0.0

Contour Center Im 0.0

Function

Residue Theorem: ∮_C f(z) dz = 2πi · Σ_{zₖ inside C} Res(f, zₖ)

The complex plane shows the function's modulus as a domain coloring. Poles appear as singularities (bright spots). The blue circular contour encloses different poles as you move/resize it. The residue theorem states that the contour integral equals 2πi times the sum of residues of poles inside the contour. Drag the contour center with sliders, or adjust the radius.