Course

Complex analysis

Study complex numbers and holomorphic functions: differentiation, contour integration, Laurent series and residues. It helps compute hard real integrals and underpins engineering, physics and number theory.

32concepts

5topics

320exercises

≈ 16 hof study approx.

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Course path

1
Complex numbers 7 concepts
Complex number · Complex conjugate · Modulus of a complex number · Polar form · Complex nth root · De Moivre's formula · Topology of the complex plane
2
Functions and limits 6 concepts
Complex function · Complex limit · Complex continuity · Complex exponential · Complex logarithm · Complex trigonometric functions
3
Analytic functions 5 concepts
Complex derivative · Cauchy-Riemann equations · Holomorphic function · Harmonic function · Möbius transformation
4
Complex integration 7 concepts
Curve and contour · Contour integral · Cauchy-Goursat theorem · Complex primitive · Cauchy integral formula · Liouville's theorem · Fundamental theorem of algebra
5
Series and residues 7 concepts
Complex Taylor series · Laurent series · Isolated singularity · Zero and pole · Residue · Residue theorem · Real integral by residues

Sources

Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003.
James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014.
Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979.