Examples#
This gallery walks through every public feature of mathematicskit.complex_analysis:
Euler’s formula, the complex plane, and the Cauchy-Riemann equations;
contour integrals, Cauchy’s integral theorem and formula, and
Liouville’s theorem; residues, Laurent series, the argument principle,
and Rouché’s theorem; essential singularities; conformal maps (Möbius
transformations, the Joukowski airfoil, and Schwarz’s lemma); and
domain coloring of complex functions.
Each script in this gallery is self-contained and can be run directly with
python examples/complex_analysis/<section>/<script>.py.
Sections#
holomorphic – Euler’s formula, the Wessel-Argand plane, and the Cauchy-Riemann equations.
contour_integrals – Cauchy’s integral theorem and integral formula, and Liouville’s theorem.
residues – the residue theorem, the argument principle, the fundamental theorem of algebra, Laurent series, and Rouché’s theorem.
series – the Casorati-Weierstrass theorem on essential singularities.
conformal_maps – Riemann’s mapping theorem via the Cayley transform, the Joukowski airfoil, Möbius transformations, and Schwarz’s lemma.
domain_coloring – phase portraits of complex functions.
Conformal maps#
Möbius transformations, the Riemann mapping theorem, and the Joukowski airfoil.
Riemann’s mapping theorem: the upper half-plane mapped conformally onto the disk
Schwarz’s lemma: self-maps of the disk fixing 0 cannot expand
Contour integrals#
Cauchy’s integral theorem and Cauchy’s integral formula.
Cauchy’s integral theorem: the integral of a holomorphic function around a closed contour is zero
Cauchy’s integral formula: boundary values determine f and every derivative
Liouville’s theorem: a bounded entire function is constant
Domain coloring#
Phase portraits: coloring each point z by the phase and modulus of f(z).
Domain coloring: reading zeros, poles, and branch cuts from a phase portrait
Holomorphic functions#
Euler’s formula and the Cauchy-Riemann equations for complex differentiability.
Euler’s formula: e^{i theta} = cos theta + i sin theta
The Cauchy-Riemann equations: u_x = v_y, u_y = -v_x
Wessel and Argand’s complex plane: multiplication rotates and scales
Residues#
The residue theorem and the argument principle.
Cauchy’s residue theorem: the contour integral is 2 pi i times the enclosed residues
The argument principle: counting zeros minus poles by winding
The fundamental theorem of algebra, by winding numbers
Laurent series: expansions with negative powers in an annulus
Series and singularities#
The behaviour of functions near isolated singularities.
The Casorati-Weierstrass theorem: near an essential singularity f comes close to every value