Advanced undergraduateAnalysisPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 39
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Optional discoveries
- Algebraic geometry Compact Riemann surfaces correspond to smooth projective complex algebraic curves.
- ← Calculus of one variable
Learning pathways
Further reading
- Lars V. Ahlfors, Complex Analysis (1953)
- Elias M. Stein and Rami Shakarchi, Complex Analysis (2003)