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Appendix B Definitions

0.1 Sets

Definition 0.1.1 Sets

Definition 0.1.2 Set equality

Definition 0.1.3 Set inclusion (subsets)

Definition 0.1.5 Set-builder notation

Definition 0.1.8 Union, intersection, difference, and complement

Definition 0.1.9 Common mathematical sets

0.2 Functions

Definition 0.2.1 Functions

Definition 0.2.5 Function equality

Definition 0.2.6 Image of a set

Definition 0.2.7 Injective, surjective, bijective

Definition 0.2.9 Function composition

Definition 0.2.10 Identity and inverse functions

0.3 Tuples and Cartesian products

Definition 0.3.1 \(n\)-tuple

Definition 0.3.4 Cartesian product (finite)

Definition 0.3.5 I-tuple

Definition 0.3.6 Cartesian product (arbitrary)

0.4 Logic

Definition 0.4.1 Logical operators

Definition 0.4.5 Logical quantifiers

1.1 Complex arithmetic

Definition 1.1.1 Complex numbers, real and imaginary parts

Convention 1.1.2 Real and complex variable names

Definition 1.1.3 Complex equality

Definition 1.1.4 Real and imaginary numbers

Definition 1.1.6 Addition, subtraction, multiplication

Convention 1.1.10 \(a+ib=a+bi\)

Definition 1.1.15 Multiplicative inverses and quotients

Definition 1.1.17 Integer powers

1.2 Geometry of complex numbers

Definition 1.2.2 Complex modulus

Convention 1.2.8 Inequality statements

Definition 1.2.10 Complex conjugation

Definition 1.2.14 Polar form

1.3 De Moivre’s formula

Definition 1.3.5 \(n\)-th roots

Definition 1.3.10 Polynomials

1.4 Topology of \(\C\)

Definition 1.4.1 Open and closed discs

Definition 1.4.3 Open sets of \(\C\)

Definition 1.4.5 Annulus

Definition 1.4.8 Closed sets

Definition 1.4.11 Boundary

Definition 1.4.14 Connected set

Definition 1.4.15 Polygonally connected

Definition 1.4.18 Extended complex plane

Definition 1.4.19 Stereographic projection

1.5 Sequences and series

Definition 1.5.1 Limit of sequence

Definition 1.5.2 Infinite limit

Definition 1.5.9 Series

1.6 Complex functions

Definition 1.6.1 Complex function

Definition 1.6.2 Real and imaginary parts

Convention 1.6.3 Real and imaginary parts

Definition 1.6.6 Complex exponential function

1.7 Logarithmic and trigonometric functions

Definition 1.7.1 Logarithm

Definition 1.7.7 Complex exponentiation

Definition 1.7.10 Trigonometric functions

Definition 1.7.12 Complex hyperbolic functions

1.8 Limits of functions and continuity

Definition 1.8.1 Limit point

Definition 1.8.3 Limit of a function

Definition 1.8.8 Continuous functions

1.9 Complex differentiation

Definition 1.9.1 Interior point

Definition 1.9.3 Complex differentiation

1.10 Cauchy-Riemann equations

Convention 1.10.1 Partial derivatives

Convention 1.10.8 Restrictions and constant functions

Definition 1.10.9 Locally constant

Definition 1.10.12 Discrete subset

1.11 Branches and harmonic functions

Definition 1.11.1 Branch

Definition 1.11.2

Definition 1.11.8 Harmonic functions

1.12 Complex paths

Definition 1.12.1 Complex path

Definition 1.12.4 Path jargon

Definition 1.12.5 Path derivative

Definition 1.12.9 Integration of complex-valued functions

1.13 Complex line integrals

Convention 1.13.1 Paths and their images

Definition 1.13.2 Complex line integral

Definition 1.13.7 Antiderivative

1.14 Cauchy-Goursat theorem

Convention 1.14.1 Paths

Convention 1.14.3 Triangles

Definition 1.14.6 Star-shaped sets

Definition 1.14.9 Elementary regions

1.16 Generalized Cauchy integral formula

Definition 1.16.6 Entire function

Definition 1.16.7 Bounded function

1.17 Complex power series

Definition 1.17.1 Complex power series

Definition 1.17.2 Power series representation

Definition 1.17.4 Sequences of functions

Definition 1.17.7 Radius of convergence

Definition 1.17.13 Analytic functions

1.19 Analytic continuation

Definition 1.19.2 Order of a zero

Definition 1.19.10 Analytic continuation

Definition 1.19.12 Real analytic

1.20 Laurent series

Definition 1.20.3 Laurent series

1.21 Cauchy residue theorem

Definition 1.21.1 Isolated singularity

Definition 1.21.2 Residue

1.22 Poles and residue computation

Definition 1.22.1 Order of isolated singularity

Definition 1.22.2 Singularity classification

1.24 Winding number

Definition 1.24.1 Winding number

Definition 1.24.4 Interior and exterior of path

Definition 1.24.9 Meromorphic

1.26 Open mapping theorem

Definition 1.26.1 Open mapping

Definition 1.26.5 Biholomorphic function

1.27 Conformal maps

Definition 1.27.1 Jacobian matrix of differentiable function

Definition 1.27.2 Conformal map

Definition 1.27.8 Möbius transformation

Definition 1.27.9 Generalized circle