Complex Function
Residues: select a function
Display Settings
Zoom (range ±R)
2.00
Grid Lines
8
Contour Circle Radius
1.00
Cursor Value
z = —
|f(z)| = —
arg f(z) = —
Re f(z) = —
Im f(z) = —
—
Number of Poles
—
Number of Zeros
—
Winding Number
—
Residue (main pole)
z-plane (Domain Coloring)
w=f(z) plane (Mapping)
|f(x)| on Real Axis and |f(iy)| on Imaginary Axis
Theory
Cauchy–Riemann equations (holomorphicity condition): $\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y},\quad \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$
Residue theorem: $\oint_C f(z)\,dz = 2\pi i \sum_k \text{Res}[f, z_k]$
Joukowski: $w = z + \dfrac{1}{z}$ (circle → airfoil)
CAE Integration: Conformal mapping in potential flow analysis (airfoil lift calculation) · Nyquist diagram in control theory (winding number = number of unstable poles) · Complex potential for electromagnetic and heat conduction fields.