Regions of the Complex Plane

Note \@{complex-regions-overlap-note} This overlaps with the definitions in Metric Spaces - I need to combine these!

An $\epsilon$ -neighborhood of a point $z_0$ is the set of all complex numbers $z$ satisfying

$\abs{z - z_0} < \epsilon.$

This is the set of all complex numbers interior to a circle of radius $\epsilon$ and center $z_0$ , but not including the circumference of the circle.

A point $z$ of a set $S$ of complex numbers is called an interior point of $S$ if there exists an $\epsilon$ -neighborhood of $z$ that contains only points of $S$ .

On the other hand, $z$ is a boundary point of $S$ if every $\epsilon$ -neighborhood of $z$ contains at least one point of $S$ and at least one point not in $S$ . The set of all boundary points of a set $S$ is called the boundary of $S$ .

A set $S$ of complex numbers is said to be open if all points in $S$ are interior points. Alternatively, a set is open if it contains none of its boundary points.

A set $S$ of complex numbers is said to be closed if it contains all of its boundary points.

Note that a set can be both open and closed! Both the empty set and $\complexes$ are both open and closed.

A set $S$ of complex numbers is said to be bounded if there exists a positive (real) number $R$ such that $\abs{z} < R$ for all $z$ in $S$ . This means that a set is bounded if all its points fit within a circle of finite radius. A set that is not bounded is said to be unbounded.

S

Definition: Domain \@{domain} A domain is an open connected set. Note that this is not the same as a domain of definition of a function. Referenced by (23 direct, 139 transitive)Direct references:Path Independentremark-12closed-path-of-path-independent-integral-is-zerotheorem-19Green's TheoremVolume IntegralDivergence Theorem of GaussStoke's TheoremDerivatives of Complex FunctionsHarmonic FunctionsElementary FunctionsContour IntegralIndepenence of Path of Contour IntegralsCauchy-Goursat TheoremCauchy's Integral TheoremCauchy Integral TheoremTaylor Expansion TheoremResidue IntegrationOuter MeasureSeparation of VariablesWell-posedness of IVPsRandom VariableParameter SpaceTransitive (depth 1):proof-of-cauchys-integral-theoremproof-of-integral-of-one-over-z-around-unit-circleParametric Familyfundamental-theorem-of-line-integralsremark-10remark-11theorem-7-intuitionA/B TestAbsolutely Continuous Random VariableBinomial DistributionCentral Limit Theoremchebyshev-fishChebyshev's Inequalitychebyshevs-inequality-noteConditional EntropyConditional Independence Given Intermediate StepConditional Mutual InformationConvolutioncorollary-88CovarianceCumulative Distribution FunctionDiscrete Random VariableEntropyExpected ValueFunction of middle variable forms Markov ChainIdentically DistributedIndependent and Identically DistributedJensen's InequalityJoint Entropyjoint-entropy-is-less-than-or-equal-to-entropy-of-partsJoint Probability DistributionLaw of Large Numberslaw-of-large-numbers-noteMarkov Chainmarkov-inequality-fishMarkov's InequalityContinuity Theorem for Moment Generating FunctionsMoment Generating Function of a SumMoment Generating FunctionMutual IndependenceMutual InformationNegative Binomial DistributionNonnegativity of Mutual InformationNormal Approximation to the BinomialNormal DistributionPairwise IndependencePoisson Approximation to the BinomialPoisson DistributionProbability Distributionproof-of-chebyshevs-inequalityproof-of-fanos-inequalityproof-of-law-of-large-numbersremark-14Reversibility of Markov ChainsSelf-InformationStandard Normal DistributionState SpaceSum of Independent Normal Random VariablesSum of Independent Poisson Random Variablestheorem-16theorem-7Uniform DistributionUniform distribution maximizes entropyVarianceTransitive (depth 2):Probability Density FunctionHypergeometric Distributionnote-25proof-of-chain-rule-for-mutual-informationGeometric DistributionProbability of ErrorConditioning Reduces EntropyCross-Entropydoubly-stochastic-maps-increase-entropy-intuitionEntropy Ratefanos-inequality-remarkGibbs' InequalityChain rule for joint entropyKL Divergencelemma-93Noiseless channel transmitting discrete symbolsnote-8proof-of-concavity-of-entropyproof-of-doubly-stochastic-maps-increase-entropyproof-of-joint-entropy-is-less-than-or-equal-to-entropy-of-partsproof-of-uniform-distribution-maximizes-entropytheorem-12Weak Asymptotic Equipartition PropertyInterpretationmarkovs-inequality-noteproof-of-markovs-inequalityproof-of-weak-asymptotic-equipartition-propertyproof-of-processing-cannot-increase-informationproof-of-corollary-95proof-of-gibbs-inequalityproof-of-lemma-93proof-of-log-sum-inequalityproof-of-conditioning-reduces-entropyConditional Relative EntropyConditionally Independentproof-of-concavity-and-convexity-of-mutual-informationErgodicnote-19proof-of-characterization-of-sufficiencyproof-of-data-processing-inequalityStationary Distribution of an Ergodic ChainStrong Asymptotic Equipartition Property (Shannon--McMillan--Breiman)Sufficient StatisticConcavity and Convexity of Mutual Informationproof-of-nonnegativity-of-mutual-informationBiasCharacterization of SufficiencyConditional Probability DistributionConvexity of Relative EntropyProduct Distributionproof-of-convexity-of-relative-entropyproof-of-nonnegativity-of-relative-entropynote-15remark-8note-3concavity-of-entropy-intuitionConfidence Interval for a Mean, Known VarianceConfidence Interval for a Mean, Unknown VarianceStandard Deviationvariance-interpretationTransitive (depth 3):proof-of-joint-is-marginal-times-conditionalproof-of-chain-rule-for-relative-entropyproof-of-nonnegativity-of-conditional-relative-entropyNonnegativity of Conditional Mutual Informationproof-of-conditional-independence-given-intermediate-stepproof-of-nonnegativity-of-conditional-mutual-informationModel Trainingproof-of-chain-rule-for-entropyproof-of-mutual-information-and-entropyconvexity-of-relative-entropy-intuitionremark-32Multiplication Rule for Distributionsproof-of-corollary-77characterization-of-sufficiency-exampleMinimal Sufficient Statistic

In the images below, a dashed line represents a boundary that is not included the in the set, while a solid line represents a boundary that is included in the set.

Squiggly blue highlighter indicates the set goes on forever and the party never ends!

Regions of the Complex Plane