Appendix C Theory
0.2 Functions
Theorem 0.2.11 Invertible is equivalent to bijective
1.1 Complex arithmetic
Theorem 1.1.9 Ring axioms
Theorem 1.1.14 Multiplicative inverses
1.2 Geometry of complex numbers
Theorem 1.2.7 Modulus properties
Theorem 1.2.12 Conjugation properties
Theorem 1.2.13 Polar form
Theorem 1.2.16 Polar form properties
1.3 De Moivre’s formula
Theorem 1.3.1 De Moivre’s formula
Theorem 1.3.8 \(n\)-th roots
Theorem 1.3.11 Elementary properties of polynomials
Theorem 1.3.12 Roots of unity
Theorem 1.3.14 Fundamental theorem of algebra
1.4 Topology of \(\C\)
Theorem 1.4.7 Open sets
Theorem 1.4.13 Open and closed via boundaries
Theorem 1.4.17 Connected sets
1.5 Sequences and series
Theorem 1.5.5 Limit properties
Theorem 1.5.10 Series properties
1.6 Complex functions
Theorem 1.6.8 Exponential function properties
1.7 Logarithmic and trigonometric functions
Theorem 1.7.4 \(\Log\) as inverse function
Theorem 1.7.11 Cosine and sine properties
Theorem 1.7.13 Complex hyperbolic functions
1.8 Limits of functions and continuity
Theorem 1.8.4 Limit of function via sequences
Theorem 1.8.6 Limits of functions
Theorem 1.8.9 Continuity and limits
Theorem 1.8.10 Continuity properties
Theorem 1.8.11 Continuous zoo
1.9 Complex differentiation
Theorem 1.9.7 Differentiable implies continuous
Theorem 1.9.8 Derivative properties
Theorem 1.9.10 Chain rule
1.10 Cauchy-Riemann equations
Theorem 1.10.2 Cauchy-Riemann equations
Theorem 1.10.5 Cauchy-Riemann converse
Corollary 1.10.6 Cauchy-Riemann on open set
Theorem 1.10.10 Functions of derivative zero
Theorem 1.10.13 Locally constant on connected sets
Theorem 1.10.14 Polar Cauchy-Riemann equations
1.11 Branches and harmonic functions
Theorem 1.11.6 Branches
Theorem 1.11.9 Harmonic conjugates
1.12 Complex paths
Theorem 1.12.8 Complex and path differentiation
Theorem 1.12.10 Complex-valued functions: fundamental theorem of calculus
Theorem 1.12.12 Complex-valued functions: modulus inequality
1.13 Complex line integrals
Theorem 1.13.5 Line integrals: fundamental theorem of calculus
Theorem 1.13.9 ML-inequality
1.14 Cauchy-Goursat theorem
Theorem 1.14.2 Antiderivative theorem
Theorem 1.14.4 Cauchy-Goursat: triangles
Theorem 1.14.8 Cauchy-Goursat: star-shaped regions
Corollary 1.14.11 Existence of log branches
Theorem 1.14.12 Elementary region building
Theorem 1.14.18 Cauchy-Goursat: boundary curve
Corollary 1.14.19 Principle of deformation
1.15 Cauchy integral formula
Theorem 1.15.2 Cauchy-integral formula
1.16 Generalized Cauchy integral formula
Theorem 1.16.1 Generalized Cauchy integral formula
Corollary 1.16.3 Antiderivatives and derivatives
Theorem 1.16.5 Morera’s theorem
Theorem 1.16.8 Liouville’s theorem
Theorem 1.16.9 Fundamental theorem of algebra
1.17 Complex power series
Theorem 1.17.5 Power series convergence
Corollary 1.17.6 Radius of convergence
Theorem 1.17.9 Power series integration and differentiation
Corollary 1.17.10 Radius of convergence
Corollary 1.17.14 Analytic implies holomorphic
Corollary 1.17.15 Uniqueness of expansions
1.18 Analyticity of holomorphic functions
Theorem 1.18.1 Holomorphic implies analytic
Theorem 1.18.6 Holomorphic equivalences
1.19 Analytic continuation
Lemma 1.19.1 Zeros of analytic functions
Theorem 1.19.6 Connectedness equivalence
Theorem 1.19.7 Limit point
Theorem 1.19.8 Rigidity of analytic functions
Corollary 1.19.11 Analytic continuation
Theorem 1.19.13 Real analytic functions
1.20 Laurent series
Theorem 1.20.1 Laurent’s theorem
1.21 Cauchy residue theorem
Theorem 1.21.4 Cauchy residue theorem
Theorem 1.21.7 Residue at simple pole
1.22 Poles and residue computation
Theorem 1.22.4 Isolated singularity
Corollary 1.22.6 Order arithmetic
1.23 Definite integrals
Theorem 1.23.1 Jordan’s inequality
Theorem 1.23.3 Fractional residue
Theorem 1.23.4 Trig integrals on \([0,2\pi]\)
1.24 Winding number
Theorem 1.24.3 Winding number properties
Theorem 1.24.7 Winding number residue theorem
Lemma 1.24.8 Argument principle
Theorem 1.24.10 Argument principle
1.25 Rouché’s theorem and open mapping theorem
Theorem 1.25.1 Rouché’s theorem
1.26 Open mapping theorem
Theorem 1.26.2 Locally \(m\)-to-one
Theorem 1.26.3 Open mapping theorem
Theorem 1.26.4 Inverse function theorem
Theorem 1.26.8 Maximum/minimum modulus theorem
Corollary 1.26.9 Schwarz’s lemma
1.27 Conformal maps
Theorem 1.27.6 Conformal maps
Theorem 1.27.11 Möbius transformations
