Universidad avanzadaAnalysisPlanificado
Complex analysis
Holomorphic functions and the Cauchy–Riemann equations, contour integration and Cauchy's theorem, power series, Liouville's theorem, the residue theorem and conformal maps.
math.analysis.complex-analysis
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 39
Aprenderás a
- Check holomorphy using the Cauchy–Riemann equations.
- Prove Cauchy's integral formula and deduce that holomorphic functions are analytic.
- Evaluate real integrals by residues, and prove the fundamental theorem of algebra from Liouville's theorem.
Definición
- f is holomorphic on an open set U ⊆ ℂ if it is complex-differentiable at every point of U.
- The residue of f at an isolated singularity a is the coefficient of (z − a)⁻¹ in its Laurent expansion about a.
Teorema
- Cauchy's integral formula.
- Holomorphic functions are infinitely differentiable and are locally equal to their Taylor series.
- Liouville: every bounded entire function is constant.
- Fundamental theorem of algebra: every non-constant complex polynomial has a complex root.
Errores frecuentes
- Assuming every real-differentiable function of two variables is holomorphic; z ↦ z̄ is not.
- Thinking a function holomorphic except at one point always extends holomorphically to that point; 1/z does not.
Descubrimientos opcionales
- Algebraic geometry Compact Riemann surfaces correspond to smooth projective complex algebraic curves.
- ← Calculus of one variable
Itinerarios de aprendizaje
Lecturas complementarias
- Lars V. Ahlfors, Complex Analysis (1953)
- Elias M. Stein and Rami Shakarchi, Complex Analysis (2003)