This is a glossary of concepts and results in real analysis and complex analysis in mathematics.
See also: list of real analysis topics, list of complex analysis topics and glossary of functional analysis.
A
[edit]- Abel
- 1. Abel sum
- 2. Abel integral
- analytic continuation
- An analytic continuation of a holomorphic function is a unique holomorphic extension of the function (on a connected open subset of ).
- argument principle
- argument principle
- Ascoli
- Ascoli's theorem says that an equicontinous bounded sequence of functions on a compact subset of has a convergent subsequence with respect to the sup norm.
B
[edit]- Borel
- 1. Borel measure.
- 2. Borel's lemma says that a given formal power series, there is a smooth function whose Taylor series coincides with the given series.
- bump
- A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function.
C
[edit]- Calderón
- Calderón–Zygmund lemma
- capacity
- Capacity of a set is a notion in potential theory.
- Carathéodory
- Carathéodory's extension theorem
- Cartan
- Cartan's theorems A and B.
- Cauchy
- 1. The Cauchy–Riemann equations are a system of differential equations such that a function satisfying it (in the distribution sense) is a holomorphic function.
- 2. Cauchy integral formula.
- 3. Cauchy residue theorem.
- 4. Cauchy's estimate.
- 5. The Cauchy principal value is, when possible, a number assigned to a function when the function is not integrable.
- 6. On a metric space, a sequence is called a Cauchy sequence if ; i.e., for each , there is an such that for all .
- Cesàro
- Cesàro summation is one way to compute a divergent series.
- contour
- The contour integral of a measurable function over a piece-wise smooth curve is .
- converge
- 1. A sequence on a topological space is said to converge to a point if for each open neighborhood of , the set is finite.
- 2. A series on a normed space (e.g., ) is said to converge if the sequence of the partial sums converges.
- convolution
- The convolution of two functions on a convex set is given by
- Cousin
- Cousin problems.
- cutoff
- cutoff function.
D
[edit]- Dedekind
- A Dedekind cut is one way to construct real numbers.
- derivative
- Given a map between normed spaces, the derivative of at a point x is a (unique) linear map such that .
- differentiable
- A map between normed space is differentiable at a point x if the derivative at x exists.
- differentiation
- Lebesgue's differentiation theorem says: for almost all x.
- Dirac
- The Dirac delta function on is a distribution (so not exactly a function) given as
- distribution
- A distribution is a type of a generalized function; precisely, it is a continuous linear functional on the space of test functions.
- divergent
- A divergent series is a series whose partial sum does not converge. For example, is divergent.
- dominated
- Lebesgue's dominated convergence theorem says converges to if is a sequence of measurable functions such that converges to pointwise and for some integrable function .
E
[edit]- edge
- Edge-of-the-wedge theorem.
- entire
- An entire function is a holomorphic function whose domain is the entire complex plane.
- equicontinuous
- A set of maps between fixed metric spaces is said to be equicontinuous if for each , there exists a such that for all with . A map is uniformly continuous if and only if is equicontinuous.
F
[edit]- Fatou
- Fatou's lemma
- Fourier
- 1. The Fourier transform of a function on is: (provided it makes sense)
G
[edit]- Gauss
- 1. The Gauss–Green formula
- 2. Gaussian kernel
- generalized
- A generalized function is an element of some function space that contains the space of ordinary (e.g., locally integrable) functions. Examples are Schwartz's distributions and Sato's hyperfunctions.
H
[edit]- Hardy
- 1. The Hardy–Littlewood maximal inequality.
- 2. Hardy space
- Hartogs
- 1. Hartogs extension theorem
- 2. Hartogs's theorem on separate holomorphicity
- harmonic
- A function is harmonic if it satisfies the Laplace equation (in the distribution sense if the function is not twice differentiable).
- Hausdorff
- The Hausdorff–Young inequality says that the Fourier transformation is a well-defined bounded operator when .
- Heaviside
- The Heaviside function is the function H on such that and .
- Hilbert space
- A Hilbert space is a real or complex inner product space that is a complete metric space with the metric induced by the inner product.
- holomorphic function
- A function defined on an open subset of is holomorphic if it is complex differentiable. Equivalently, a function is holomorphic if it satisfies the Cauchy–Riemann equations (in the distribution sense if the function is not differentiable).
I
[edit]- integrable
- A measurable function is said to be integrable if .
- integral
- 1. The integral of the indicator function on a measurable set is the measure (volume) of the set.
- 2. The integral of a measurable function is then defined by approximating the function by linear combinations of indicator functions.
L
[edit]- Lebesgue integral
- Lebesgue integral.
- Lebesgue measure
- Lebesgue measure.
- Lelong
- Lelong number.
- Levi
- Levi's problem asks to show a pseudoconvex set is a domain of holomorphy.
- line integral
- Line integral.
- Liouville
- Liouville's theorem says a bounded entire function is a constant function.
- Lipschitz
- 1. A map between metric spaces is said to be Lipschitz continuous if .
- 2. A map is locally Lipschitz continuous if it is Lipschitz continuous on each compact subset.
M
[edit]- maximum
- The maximum principle says that a maximum value of a harmonic function in a connected open set is attained on the boundary.
- measurable function
- A measurable function is a structure-preserving function between measurable spaces in the sense that the preimage of any measurable set is measurable.
- measurable set
- A measurable set is an element of a σ-algebra.
- measurable space
- A measurable space consists of a set and a σ-algebra on that set which specifies what sets are measurable.
- measure
- A measure is a function on a measurable space that assigns to each measurable set a number representing its measure or size. Specifically, if X is a set and Σ is a σ-algebra on X, then a set-function μ from Σ to the extended real number line is called a measure if the following conditions hold:
- Non-negativity: For all
- Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ,
- measure space
- A measure space consists of a measurable space and a measure on that measurable space.
- meromorphic
- A meromorphic function is an equivalence class of functions that are locally fractions of holomorphic functions.
- method of stationary phase
- The method of stationary phase.
- microlocal
- The notion microlocal refers to a consideration on the cotangent bundle to a space as opposed to that on the space itself. Explicitly, it amounts to considering functions on both points and momenta; not just functions on points.
- Morera
- Morera's theorem says a function is holomorphic if the integrations of it over arbitrary closed loops are zero.
- Morse
- Morse function.
N
[edit]- Nevanlinna theory
- Nevanlinna theory concerns meromorphic functions.
- net
- A net is a generalization of a sequence.
- normed vector space
- A normed vector space, also called a normed space, is a real or complex vector space V on which a norm is defined. A norm is a map satisfying four axioms:
- Non-negativity: for every ,.
- Positive definiteness: for every , if and only if is the zero vector.
- Absolute homogeneity: for every scalar and ,
- Triangle inequality: for every and ,
O
[edit]- Oka
- Oka's coherence theorem says the sheaf of holomorphic functions is coherent.
- open
- The open mapping theorem (complex analysis)
- oscillatory integral
- An oscillatory integral can give a sense to a formal integral expression like
P
[edit]- Paley
- Paley–Wiener theorem
- phase
- The phase space to a configuration space (in classical mechanics) is the cotangent bundle to .
- plurisubharmonic
- A function on an open subset is said to be plurisubharmonic if is subharmonic for in a neighborhood of zero in and points in .
- Poisson
- Poisson kernel
- power series
- A power series is informally a polynomial of infinite degree; i.e., .
- pseudoconex
- A pseudoconvex set is a generalization of a convex set.
R
[edit]- real-analytic
- A real-analytic function is a function given by a convergent power series.
- Riemann
- 1. The Riemann integral of a function is either the upper Riemann sum or the lower Riemann sum when the two sums agree.
- 2. The Riemann zeta function is a (unique) analytic continuation of the function (it's more traditional to write for ).
- 3. The Riemann hypothesis, still a conjecture, says each nontrivial zero of the Riemann zeta function has real part equal to .
- Runge
- 1. Runge's approximation theorem.
- 2. Runge domain.
S
[edit]- Sato
- Sato's hyperfunction, a type of a generalized function.
- Schwarz
- A Schwarz function is a function that is both smooth and rapid-decay.
- semianalytic
- The notion of semianalytic is an analog of semialgebraic.
- sequence
- A sequence on a set is a map .
- series
- A series is informally an infinite summation process . Thus, mathematically, specifying a series is the same as specifying the sequence of the terms in the series. The difference is that, when considering a series, one is often interested in whether the sequence of partial sums converges or not and if so, to what.
- σ-algebra
- A σ-algebra on a set is a nonempty collection of subsets closed under complements, countable unions, and countable intersections.
- Stieltjes
- Stieltjes–Vitali theorem
- subanalytic
- subanalytic.
- subharmonic
- A twice continuously differentiable function is said to be subharmonic if where is the Laplacian. The subharmonicity for a more general function is defined by a limiting process.
- subsequence
- A subsequence of a sequence is another sequence contained in the sequence; more precisely, it is a composition where is a strictly increasing injection and is the given sequence.
- support
- 1. The support of a function is the closure of the set of points where the function does not vanish.
- 2. The support of a distribution is the support of it in the sense in sheaf theory.
T
[edit]- Tauberian
- Tauberian theory is a set of results (called tauberian theorems) concerning a divergent series; they are sort of converses to abelian theorems but with some additional conditions.
- Taylor
- Taylor expansion
- tempered
- A tempered distribution is a distribution that extends to a continuous linear functional on the space of Schwarz functions.
- test
- A test function is a compactly-supported smooth function.
U
[edit]- uniform
- 1. A sequence of maps from a topological space to a normed space is said to converge uniformly to if .
- 2. A map between metric spaces is said to be uniformly continuous if for each , there exist a such that for all with .
V
[edit]- Vitali
- Vitali covering lemma.
W
[edit]- Weierstrass
- 1. Weierstrass preparation theorem.
- 2. Weierstrass M-test.
- Weyl
- 1. Weyl calculus.
- 2. Weyl quantization.
- Whitney
- 1. The Whitney extension theorem gives a necessary and sufficient condition for a function to be extended from a closed set to a smooth function on the ambient space.
- 2. Whitney stratification
References
[edit]- Grauert, Hans; Remmert, Reinhold (1984). Coherent Analytic Sheaves. Springer.
- Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 978-0-387-90088-9, MR 0033869, Zbl 0283.28001
- Hörmander, Lars (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.
- Hörmander, Lars (1966). An Introduction to Complex Analysis in Several Variables. Van Nostrand.
- Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358.
- Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill. ISBN 978-0-07-054234-1.
Further reading
[edit]- Semiclassical Microlocal Analysis(2020 Fall) by 王作勤 (wangzuoq)