Complex Analysis
Year 2 · 6 sections · 59 concepts · 136 practice questions
What's covered
free1 · Holomorphic functions9 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Complex differentiability and holomorphic functions
- Proposition 1.2 · Rules for complex derivatives
- Theorem 1.3 · The Cauchy–Riemann equations are necessary
- Theorem 1.4 · Cauchy–Riemann with continuous partials implies holomorphic
- Corollary 1.5 · Real and imaginary parts are harmonic
- Example 1.6 · Finding a harmonic conjugate
- Theorem 1.7 · Power series are holomorphic
- Check 1.8 · Check: is this function holomorphic?
2 · Elementary functions11 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · The complex exponential and trigonometric functions
- Proposition 2.2 · Properties of the exponential
- Definition 2.3 · Logarithms and branches
- Proposition 2.4 · The principal logarithm is holomorphic off the branch cut
- Definition 2.5 · Complex powers
- Example 2.6 · Branches of the square root
- Definition 2.7 · Möbius transformations
- Proposition 2.8 · Möbius transformations form a group
- Theorem 2.9 · Möbius maps send circles and lines to circles and lines
- Check 2.10 · Check: logs, powers and Möbius maps
3 · Contour integration9 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Contours and contour integrals
- Example 3.2 · Integrating powers around a circle
- Proposition 3.3 · The estimation (ML) lemma
- Theorem 3.4 · Antiderivatives and path independence
- Theorem 3.5 · Cauchy's theorem for triangles (Goursat)
- Theorem 3.6 · Cauchy's theorem on star-shaped domains
- Corollary 3.7 · Deforming contours
- Check 3.8 · Check: evaluating contour integrals
4 · Cauchy's integral formula11 concepts
- Overview 4 · The plan of this section
- Theorem 4.1 · Cauchy's integral formula
- Theorem 4.2 · Holomorphic functions are analytic
- Corollary 4.3 · Infinitely differentiable with the derivative formula
- Proposition 4.4 · Cauchy's estimates
- Theorem 4.5 · Liouville's theorem
- Corollary 4.6 · The fundamental theorem of algebra
- Theorem 4.7 · Morera's theorem
- Theorem 4.8 · The identity theorem and isolated zeros
- Theorem 4.9 · Maximum modulus principle
- Check 4.10 · Check: using Cauchy's integral formula
5 · Series and singularities9 concepts
- Overview 5 · The plan of this section
- Theorem 5.1 · Laurent series
- Example 5.2 · Computing Laurent series from known series
- Definition 5.3 · Isolated singularities
- Theorem 5.4 · Riemann's removable singularity theorem
- Proposition 5.5 · Characterising poles
- Theorem 5.6 · Casorati–Weierstrass theorem
- Definition 5.7 · Meromorphic functions and the point at infinity
- Check 5.8 · Check: classifying a singularity
6 · Residues10 concepts
- Overview 6 · The plan of this section
- Definition 6.1 · Winding number
- Theorem 6.2 · The residue theorem
- Proposition 6.3 · Computing residues at poles
- Example 6.4 · Trigonometric integrals over a period
- Proposition 6.5 · Integrals of rational functions over the real line
- Proposition 6.6 · Jordan's lemma and Fourier-type integrals
- Theorem 6.7 · The argument principle
- Theorem 6.8 · Rouché's theorem
- Check 6.9 · Check: evaluating integrals by residues
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Proposition 1.2 · Rules for complex derivatives
All the familiar rules hold; polynomials entire, rational functions holomorphic off their poles.
Proof
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