Appendix E Examples
0.1 Sets
0.2 Functions
Example 0.2.3 Arithmetic operations as functions
Example 0.2.8 Role of domain and codomain in injectivity and surjectivity
0.4 Logic
Example 0.4.7 Modeling “Every positive number has a square-root”
Example 0.4.10 The limit does not exist
0.5 Proof techniques
Example 0.5.3 Proof: invertible is equivalent to bijective
Example 0.5.4 Proof by contradiction
Example 0.5.8 Weak induction
Example 0.5.10 Strong induction
1.1 Complex arithmetic
Example 1.1.7 Complex arithmetic
Example 1.1.12 Real square roots
Example 1.1.16 Inverses and quotients
1.2 Geometry of complex numbers
Example 1.2.6 Circles and discs
Example 1.2.15 Polar form
Example 1.2.18 Polar form arithmetic
1.3 De Moivre’s formula
Example 1.3.2 De Moivre’s formula
Example 1.3.3 Double-angle formulas
Example 1.3.4 Triple-angle formula
Example 1.3.6 Cube-roots of \(8i\)
Example 1.3.13 Factoring \(f(x)=x^6-1\)
1.4 Topology of \(\C\)
Example 1.4.4 Open sets
Example 1.4.9 Open, closed, neither
Example 1.4.12 Boundary
1.5 Sequences and series
Example 1.5.6 Limit of sequence
Example 1.5.7 Divergent sequence
Example 1.5.8 Sequence with infinite limit
Example 1.5.11 Geometric series
1.6 Complex functions
Example 1.6.5 Real and imaginary parts
Example 1.6.10 Squaring transformation
Example 1.6.13 Exponential transformation
1.7 Logarithmic and trigonometric functions
Example 1.7.5 Logarithm
Example 1.7.6 Invalid identity
Example 1.7.8 Powers
Example 1.7.9 \(n\)-th roots
1.8 Limits of functions and continuity
Example 1.8.2 Limit points
Example 1.8.5 No limit
Example 1.8.7 Limit of power function
1.9 Complex differentiation
Example 1.9.6 Elementary examples
Example 1.9.9 Polynomials
1.10 Cauchy-Riemann equations
Example 1.10.3 Cauchy-Riemann verification
Example 1.10.4 Complex conjugation
Example 1.10.7 Exponential and trigonometric functions
Example 1.10.15 Derivative of \(\Log\)
Example 1.10.16 Reciprocal function
Example 1.10.17 Derivative of power function
1.11 Branches and harmonic functions
Example 1.11.4 Relating \(\alpha\)-cut branches
Example 1.11.5 \(\Log_\alpha(z)\) is holomorphic
Example 1.11.7 Branches of cube-root function
Example 1.11.10 Harmonic conjugates
1.12 Complex paths
Example 1.12.3 Parametrized curves
Example 1.12.7 Tangent vectors
Example 1.12.11 Complex-valued function integration
1.13 Complex line integrals
Example 1.13.4 Classic line integral
Example 1.13.6 Line integral: FTC
1.14 Cauchy-Goursat theorem
Example 1.14.13 Star-shaped and elementary regions
Example 1.14.21 Principle of deformation
1.15 Cauchy integral formula
Example 1.15.3 Cauchy Integral formula
Example 1.15.5 Cauchy integral formula: cosine
1.16 Generalized Cauchy integral formula
Example 1.16.4 Generalized Cauchy integral formula
1.17 Complex power series
Example 1.17.3 Power series representation
Example 1.17.8 Expansions and radii of convergence
Example 1.17.11 Differentiating power series
Example 1.17.12 Power series for \(\Log\)
1.19 Analytic continuation
Example 1.19.3 Order of a zero
Example 1.19.14 Real analytic functions
1.20 Laurent series
Example 1.20.4 Laurent series expansion
Example 1.20.6 Different Laurent series
Example 1.20.8 Truncated Laurent series
Example 1.20.10 Laurent series of rational function
1.21 Cauchy residue theorem
Example 1.21.5 Cauchy residue theorem
Example 1.21.6 Cauchy residue theorem: two singularities
Example 1.21.8 Residue at simple pole
1.22 Poles and residue computation
Example 1.22.3 Singularity classification
Example 1.22.8 Residue computation
1.23 Definite integrals
Example 1.23.5 Trig integral over \([0,2\pi]\)
Example 1.23.7 Rational function on real line
Example 1.23.8 Trig integral over real line
Example 1.23.10 Logarithmic improper integral
Example 1.23.11 Sinc function
Example 1.23.13 Integral of power function
1.24 Winding number
Example 1.24.2 Winding numbers of circles
1.25 Rouché’s theorem and open mapping theorem
Example 1.25.2 Roots of polynomials
1.27 Conformal maps
Example 1.27.7 Conformal map: squaring function
Example 1.27.12 Möbius transformation
