Theorems in analysis

Theorems in analysis refer to significant results within the field of mathematical analysis, which studies limits, continuity, differentiation, integration, sequences, series, and function properties, among other topics. Analysis is a branch of mathematics that is foundational for understanding calculus and many other areas of mathematics.

Convergence tests are mathematical techniques used to determine whether a series or sequence converges (approaches a finite limit) or diverges (grows indefinitely or does not settle at any finite value). These tests are particularly important in the study of infinite series in calculus and analysis, as they help evaluate the behavior of sums of infinitely many terms.

Abel's test is a convergence test for series of the form \(\sum a_n b_n\), where \(a_n\) is a monotonic sequence that converges to 0, and \(\{b_n\}\) is a bounded sequence. The test is useful in determining the convergence of series when you are dealing with products of sequences.

The Alternating Series Test is a method used to determine the convergence of an alternating series. An alternating series is a series of the form: \[ \sum_{n=1}^{\infty} (-1)^{n} a_n \] where \( a_n \) is a sequence of positive terms. The test provides a way to conclude that the series converges under specific conditions.

Cauchy's convergence test, also known as the Cauchy criterion for convergence, is a method used to determine whether a sequence of real or complex numbers converges. This criterion is particularly useful because it provides a way to check for convergence without requiring knowledge of the limit to which the sequence converges.

Cauchy's limit theorem is a result in real analysis that provides conditions under which a sequence of real or complex numbers converges. Specifically, it deals with the concepts of convergence in the context of sequences.

The Cauchy condensation test is a convergence test used in the analysis of infinite series, particularly for series with non-negative terms. It provides a method to determine the convergence or divergence of a series based on the behavior of a related series.

The Dini test is a method used in mathematics, particularly in real analysis, to determine the convergence of a sequence of functions. More specifically, it is applicable to the study of pointwise convergence of a sequence of real-valued functions defined on a common domain. The test is based on the idea of comparing the behavior of the functions in the sequence with a "monotone" function or to establish some control over their convergence through the use of integrals.

The Direct Comparison Test is a method used in calculus to determine the convergence or divergence of an infinite series. It compares the series in question with a known benchmark series whose convergence behavior is already established. This test is particularly useful when dealing with series that have positive terms.

Dirichlet's test is a convergence test used primarily to determine the convergence of certain types of infinite series. It is particularly useful for series that may contain oscillatory components. The test is named after the German mathematician Johann Peter Gustav Lejeune Dirichlet.

The Integral Test for convergence is a method used to determine whether a series converges or diverges by comparing it to an improper integral. It applies specifically to series that consist of positive, decreasing functions. ### Statement of the Integral Test Let \( f(x) \) be a positive, continuous, and decreasing function for \( x \geq N \) (where \( N \) is some positive integer).

The Limit Comparison Test is a method used in calculus to determine the convergence or divergence of infinite series. It is particularly useful when the series in question is difficult to analyze directly. The test compares the given series with a known benchmark series.

The Nth-term test, also known as the Divergence Test, is a method used in calculus and series analysis to determine the convergence or divergence of an infinite series. It specifically applies to a series of the form: \[ \sum_{n=1}^{\infty} a_n \] where \(a_n\) is the nth-term of the series.

The Ratio Test is a method in mathematical analysis, particularly useful for determining the convergence or divergence of infinite series. It is often used for series whose terms involve factorials, exponentials, or other functions where the terms can grow rapidly. ### Statement of the Ratio Test Let \( \{a_n\} \) be a sequence of positive terms.

The Root Test is a method used to determine the convergence or divergence of an infinite series. Specifically, it helps assess the behavior of a series of the form: \[ \sum_{n=1}^{\infty} a_n \] where \( a_n \) is a sequence of real or complex numbers. The primary approach is based on the concept of the \( n \)-th root of the absolute value of the terms in the series.

The Stolz–Cesàro theorem is a result in real analysis that provides a method for evaluating limits of sequences, particularly those that appear in the context of quotients of two sequences. It is often used in situations where the conventional techniques for finding limits may not be easily applicable.

Fixed-point theorems are fundamental results in mathematics that establish conditions under which a function will have a point that maps to itself. In simpler terms, if you have a function \( f \) defined on a certain space, a fixed point \( x \) satisfies the equation \( f(x) = x \). Fixed-point theorems are widely applicable in various areas such as analysis, topology, and applied mathematics.

The Atiyah–Bott fixed-point theorem is a fundamental result in algebraic topology and differential geometry, developed by mathematicians Michael Atiyah and Raoul Bott in the context of the study of fixed points of smooth maps on manifolds.

Bekić's theorem is a result in the field of functional analysis, specifically concerning the properties of certain types of topological vector spaces. The theorem addresses the conditions under which a set of continuous linear functionals on a topological vector space can separate points in the space.

The Borel fixed-point theorem is a result in topology, particularly in the context of more general spaces than just traditional fixed-point theorems. It states that any continuous function from a compact convex set in a finite-dimensional Euclidean space to itself has at least one fixed point.

The Brouwer Fixed-Point Theorem is a fundamental result in topology, specifically in the field of fixed-point theory. It states that any continuous function mapping a compact convex set to itself has at least one fixed point.

The Browder Fixed-Point Theorem is a result in functional analysis and topological fixed-point theory, named after the mathematician Felix Browder. This theorem extends the classical Brouwer Fixed-Point Theorem to more general contexts, particularly in infinite-dimensional spaces.

The Earle–Hamilton fixed-point theorem is a result in the field of topology, particularly in the study of fixed points in continuous functions.

A fixed-point theorem is a fundamental result in various branches of mathematics, particularly in analysis and topology, that asserts the existence of fixed points under certain conditions. A fixed point of a function is a point that is mapped to itself by the function. Formally, if \( f: X \rightarrow X \) is a function on a set \( X \), then a point \( x \in X \) is a fixed point if \( f(x) = x \).

Fixed-point theorems are fundamental results in mathematics that guarantee the existence of points that remain unchanged under certain mappings. While fixed-point theorems are traditionally studied in finite-dimensional spaces (like the well-known Banach and Brouwer Fixed-Point Theorems), their generalization to infinite-dimensional spaces presents some unique challenges and requires different techniques. Here’s an overview of some of the key concepts and results related to fixed-point theorems in infinite-dimensional spaces: ### 1.

The Kleene fixed-point theorem is a fundamental result in theoretical computer science and mathematical logic, particularly in the context of domain theory and functional programming. Named after Stephen Cole Kleene, it provides a framework for understanding the existence of fixed points in certain types of functions. In simple terms, a fixed point of a function \( f \) is a value \( x \) such that \( f(x) = x \).

The Lefschetz Fixed-Point Theorem is a fundamental result in algebraic topology that provides a criterion for determining the existence of fixed points of continuous maps on topological spaces. It is particularly useful when dealing with maps between compact, connected, and oriented manifolds.

The Markov–Kakutani fixed-point theorem is a generalization of the classical Brouwer fixed-point theorem, designed for multi-valued functions (or correspondences). It is important in various areas such as game theory, economics, and optimization.

Nielsen theory, often associated with the work of mathematician and physicist Nielsen, primarily pertains to the field of topological and algebraic invariants in the context of knot theory and three-manifolds. One of the key contributions of Nielsen is his work on the concept of "Nielsen classes," which relate to the classification of covering spaces of surfaces and the study of fundamental groups.

The Ryll-Nardzewski fixed-point theorem is a result in the field of functional analysis, specifically concerning fixed points in nonatomic convex sets in topological vector spaces. It generalizes certain fixed-point results, including the well-known Brouwer fixed-point theorem, to more general settings.

The Schauder fixed-point theorem is a fundamental result in fixed-point theory, particularly in the context of functional analysis and topology. It provides conditions under which a continuous function mapping a convex compact subset of a Banach space (or more generally, in a topological vector space) has at least one fixed point.

The Single-Crossing Condition (SCC) is a concept used primarily in economics, particularly in the context of auction theory, mechanism design, and social choice theory. It refers to a specific property of preference orderings among different agents or individuals regarding a set of alternatives. Under the Single-Crossing Condition, the preference rankings of the individuals (or types) can only cross at most once when plotted against a single dimension of preference.

In mathematical analysis and other fields of mathematics, a "lemma" is a preliminary proposition or statement that is proven to aid in the proof of a larger theorem. The term "lemma" comes from the Greek word "lemma," which means "that which is received" or "that which is taken." In effect, results that are designated as lemmas are often foundational results that help establish more complex results.

Auerbach's lemma is a result in functional analysis, specifically in the area concerning the geometry of Banach spaces. It is often used in the study of dual spaces and the properties of linear functionals. The lemma essentially characterizes the relationship between a subspace of a Banach space and its dual.

The Bramble–Hilbert lemma is a result in the mathematical field of numerical analysis and finite element methods. It provides a fundamental estimate that is crucial in the approximation properties of finite element spaces, particularly in the context of solving partial differential equations.

The Calderón–Zygmund lemma is a fundamental result in the theory of singular integrals and is often used in various areas of analysis, including harmonic analysis and partial differential equations. It is named after the mathematicians Alberto Calderón and Anton Zygmund, who made significant contributions to the field.

In differential geometry and calculus, the concepts of closed and exact differential forms are crucial for understanding forms on manifolds, specifically in the context of integration and topology.

Céa's lemma is a result in the field of functional analysis and calculus of variations, particularly in the context of optimal control problems. The lemma is often used to derive estimates for the behavior of solutions to variational problems. In a general sense, Céa's lemma states that under certain conditions, the error in the approximation of a functional can be controlled in terms of the norm of a corresponding linear functional applied to the error of the function.

The Estimation Lemma is a concept used in mathematical analysis, particularly in the context of sequences and series. It is often associated with the estimation of the behavior of functions or sequences under certain conditions. While the term "Estimation Lemma" may not refer to a single, universally recognized theorem, it typically involves methods to estimate bounds or behavior for sequences, series, or integrals.

Grönwall's inequality is an important result in the field of differential equations and analysis, particularly useful for establishing the existence and uniqueness of solutions to differential equations. It provides a way to estimate functions that satisfy certain integral inequalities. There are two common forms of Grönwall's inequality: the integral form and the differential form.

Jordan's Lemma is a result in complex analysis that is particularly useful in evaluating certain types of integrals involving oscillatory functions over infinite intervals. It provides a method for showing that specific integrals vanish under certain conditions, especially when the integrands involve exponential factors.

Lebesgue's lemma is a result in measure theory related to the behavior of measurable functions and their integrals.

Lions–Magenes lemma is a result in the field of functional analysis, particularly in the context of Sobolev spaces and partial differential equations. It provides a crucial tool for establishing the regularity and control of solutions to elliptic and parabolic differential equations. The lemma is typically used to handle boundary value problems, allowing one to obtain estimates of solutions in various norms, which is essential for understanding the existence and uniqueness of solutions as well as their continuity and differentiability properties.

Mazur's Lemma is a result in functional analysis, specifically in the context of convex analysis and the study of weak compactness in Banach spaces. The lemma is named after the Polish mathematician Stanisław Mazur. The central idea of Mazur's Lemma concerns weakly convergent sequences and the nature of the weak closure of convex sets.

The Poincaré lemma is a fundamental result in differential geometry and algebraic topology that pertains to the properties of differential forms on a differentiable manifold.

The Schwarz lemma is a fundamental result in complex analysis that provides important insights into the behavior of holomorphic functions. Specifically, it applies to holomorphic functions defined on the unit disk (the set of complex numbers whose modulus is less than 1).

Spijker's lemma is a result in functional analysis, specifically dealing with the properties of bounded linear operators on Banach spaces. The lemma provides conditions under which certain sequences of bounded linear operators exhibit specific convergence properties. While Spijker's lemma does not have one widely acknowledged statement applicable in all contexts, it typically relates to convergence properties in the context of compact operators or the spectral theory of linear operators.

The Stewart–Walker lemma is a result in the field of differential geometry, particularly in the study of Riemannian manifolds. It is specifically related to the curvature of manifolds and provides conditions under which the curvature tensor can be expressed in terms of the metric tensor and its derivatives. The lemma is often invoked in the context of proving properties about space forms and the relationship between curvature and geometric structures on manifolds.

Weyl's lemma is a result in the theory of partial differential equations, particularly concerning solutions to the Laplace equation. The lemma states that if a function \( u \) is harmonic (i.e.

Wiener's lemma is a result in functional analysis and harmonic analysis, particularly related to the theory of Fourier series and the spaces of functions. It is named after Norbert Wiener, who contributed significantly to the field.

Analytic number theory is a branch of number theory that uses techniques from mathematical analysis to solve problems about integers and prime numbers. Several important theorems form the foundation of this field. Here are some of the prominent theorems and concepts within analytic number theory: 1. **Prime Number Theorem**: This fundamental theorem describes the asymptotic distribution of prime numbers.

The Barban–Davenport–Halberstam theorem is a result in number theory, specifically in the area of additive number theory and the distribution of prime numbers. It provides a way to estimate the size of the prime numbers in certain ranges. More formally, the theorem deals with the distribution of prime numbers in arithmetic progressions and gives a bound on the number of primes in intervals of certain lengths.

The Brun–Titchmarsh theorem is a result in analytic number theory that provides an asymptotic estimate for the number of primes in an arithmetic progression. More specifically, it addresses the distribution of prime numbers in the form \( a + nd \), where \( a \) and \( d \) are coprime integers, and \( n \) ranges over the natural numbers.

Chen's theorem is a result in number theory, specifically in the area of prime numbers. It states that every sufficiently large even integer can be expressed as the sum of a prime and the product of at most two primes. The theorem can be seen as a refinement of the Goldbach conjecture, which posits that every even integer greater than 2 can be expressed as the sum of two primes.

The Friedlander–Iwaniec theorem is a result in number theory, specifically in the area of additive number theory concerning the distribution of prime numbers. It was established by the mathematicians J. Friedlander and H. Iwaniec in the early 1990s.

The Hardy–Ramanujan theorem, also known as the Hardy-Ramanujan asymptotic formula, describes the asymptotic behavior of the partition function \( p(n) \), which counts the number of ways to express a positive integer \( n \) as a sum of positive integers, disregarding the order of the summands.

The Kronecker limit formula is an important result in the theory of modular forms and number theory. It relates the behavior of certain L-functions to the special values of those functions at integers. Specifically, it provides a way to compute the special value of an L-function associated with a point on a certain modular curve. The formula can be stated in the context of the Dedekind eta function and the Eisenstein series.

The Landau prime ideal theorem is a result in the field of algebra, specifically in commutative algebra and the theory of rings. It concerns the structure of prime ideals in a non-zero commutative ring.

The Landsberg–Schaar relation is a concept in the field of thermodynamics, particularly in relation to the thermoelectric properties of materials. It establishes a relationship between the electrical conductivity, the Seebeck coefficient, and the thermal conductivity of a material. This relation is significant because it helps to optimize materials for thermoelectric applications, such as in power generation or cooling devices.

Linnik's theorem is a result in number theory that pertains to the distribution of prime numbers in arithmetic progressions. Specifically, it concerns the distribution of primes in progressions of the form \( a \mod q \) where \( a \) and \( q \) are coprime integers.

Maier's theorem is a result in number theory related to the distribution of prime numbers. Specifically, it deals with the existence of certain arithmetic progressions among prime numbers. The theorem is typically discussed in the context of additive number theory and is named after the mathematician Helmut Maier, who contributed to the understanding of the distribution of primes.

The Petersson trace formula is an important result in the theory of modular forms and number theory. It provides a relationship between the eigenvalues of Hecke operators on modular forms and the values of L-functions at certain critical points. The formula is named after the mathematician Heinrich Petersson, who was instrumental in its development. In its most common form, the Petersson trace formula connects the spectral theory of automorphic forms with the arithmetic of numbers through the Fourier coefficients of modular forms.

The Prime Number Theorem (PNT) is a fundamental result in number theory that describes the asymptotic distribution of prime numbers. It states that the number of prime numbers less than a given number \( n \), denoted as \( \pi(n) \), is approximately equal to \( \frac{n}{\log(n)} \), where \( \log(n) \) is the natural logarithm of \( n \).

Ramanujan's Master Theorem is a result from the theory of infinite series and analytic functions, developed by the Indian mathematician Srinivasa Ramanujan. It provides a way to evaluate certain types of series involving powers of \( n \) and can be used to find sums of generating functions.

The Riemann-von Mangoldt formula is an important result in analytic number theory that provides an asymptotic expression for the number of prime numbers less than or equal to a certain value \( x \). More formally, it relates the distribution of prime numbers to the Riemann zeta function, a central object of study in number theory.

The Siegel–Walfisz theorem is a result in analytic number theory that provides a relationship between the distribution of prime numbers and certain arithmetic functions. Specifically, it deals with the distribution of prime numbers in arithmetic progressions and offers an asymptotic formula for the count of such primes.

Vinogradov's mean-value theorem is a result in additive number theory that concerns the distribution of the values of additive functions. It has significant implications for the study of Diophantine equations and is particularly important in the field of analytic number theory. The theorem essentially states that for a certain class of additive functions (typically of the type that can be exhibited as sums of integers), the average number of representations of a number as a sum of other integers can be understood in a mean-value sense.

In approximation theory, several theorems provide fundamental insights into how functions can be approximated by simpler functions, such as polynomials, trigonometric series, or other basis functions. Here are some key theorems and concepts in approximation theory: 1. **Weierstrass Approximation Theorem**: This theorem states that any continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

Arakelyan's theorem is a result in approximation theory, particularly concerning the approximation of continuous functions by certain classes of functions. It states that if \( f \) is a continuous function defined on a compact subset of \( \mathbb{R}^n \) that is not identically zero, then there exists a sequence of functions that can approximate \( f \) arbitrarily closely.

The Erdős–Turán inequality is a result in combinatorial number theory that deals with the distribution of sums in sequences of integers.

Fejér's theorem is a result in the theory of Fourier series, specifically concerning the convergence of the Fourier series of a periodic function. It states that if \( f \) is a piecewise continuous function on the interval \([-L, L]\), then the sequence of partial sums of its Fourier series converges uniformly to the average of the left-hand and right-hand limits of \( f \) at each point.

The Lethargy Theorem, also known as the Lethargy Principle, is a concept from the field of probability theory, often discussed in the context of computer simulations and the analysis of stochastic processes. Specifically, it deals with the tendencies of certain stochastic systems to become less responsive or "lethargic" over time under particular conditions.

Mergelyan's theorem is a result in complex analysis concerning the approximation of holomorphic functions (functions that are complex differentiable) on compact subsets of complex domains. Specifically, it deals with the approximation of functions by polynomials.

The Walsh–Lebesgue theorem is a result in the field of harmonic analysis and real analysis concerning the properties of functions represented by Walsh series, which are expansions using Walsh functions. Walsh functions are a specific orthonormal basis used in the space of square-integrable functions on the interval [0, 1].

Wirtinger's representation theorem and projection theorem are fundamental results in mathematical analysis, particularly in the fields of functional analysis and the theory of Sobolev spaces. They are often applied in the study of harmonic functions, the solution of partial differential equations, and variational problems. ### Wirtinger's Representation Theorem: The Wirtinger representation theorem provides a way to connect the Dirichlet energy of functions to their boundary conditions.

The ATS theorem, also known as the Aharonov–Bohm theorem, is a fundamental result in quantum mechanics that illustrates the importance of electromagnetic potentials in the behavior of charged particles, even in regions where the electric and magnetic fields are zero.

Ahlfors' finiteness theorem is a result in complex analysis and several complex variables, particularly in the study of Riemann surfaces and compact complex manifolds. The theorem is named after Lars Ahlfors, a prominent mathematician known for his contributions to complex analysis. The theorem essentially states that for a compact Riemann surface (or a compact complex manifold), the number of non-constant meromorphic functions (or rational functions) that can be defined on it is finite.

The Ahlfors measure conjecture is a conjecture in the field of complex analysis and geometric function theory, specifically relating to quasiconformal mappings and the properties of certain topological spaces. Named after the mathematician Lars Ahlfors, this conjecture deals with the existence of a specific type of measure associated with quasiconformal mappings.

The Atkinson-Mingarelli theorem is a result in the field of differential equations, particularly in the context of boundary value problems for second-order ordinary differential equations (ODEs). It deals with the existence of multiple solutions to certain types of boundary value problems. The theorem essentially states conditions under which a second-order linear differential equation can have multiple solutions based on its boundary conditions and the nature of the functions involved.

The Babuška–Lax–Milgram theorem is a result in functional analysis and the theory of partial differential equations (PDEs), particularly concerning the solvability of boundary value problems. It is named after mathematicians Ivo Babuška, Gilbert Lax, and Alexander Milgram, who contributed to its development. The theorem provides conditions under which a linear operator associated with a boundary value problem possesses a unique solution and characterizes this solution in terms of bounded linear functionals.

The Besicovitch covering theorem is a result in measure theory and geometric measure theory that deals with the covering of sets in Euclidean space by balls. It is particularly important in the context of studying properties of sets of points in \(\mathbb{R}^n\) and has applications in various areas such as size theory, geometric measure theory, and analysis.

The Brezis–Gallouët inequality is an important result in functional analysis and partial differential equations, particularly in the context of Sobolev spaces. It provides a bound for a certain type of functional involving the fractional Sobolev norms. Specifically, the inequality can be stated as follows: Let \( n \geq 1 \) and \( p \in (1, n) \).

Carathéodory's existence theorem is a fundamental result in the theory of ordinary differential equations (ODEs). It provides conditions under which a first-order ordinary differential equation has at least one solution. The theorem is particularly important for equations that may not have Lipschitz continuity, allowing for broader applications.

The Cartan–Kuranishi prolongation theorem is a result in the field of differential geometry and the theory of differential equations, particularly in relation to the existence of local solutions to differential equations and the structures of their solutions. The theorem is attributed to the work of Henri Cartan and Masao Kuranishi, who contributed fundamentally to the understanding of deformation theory and the theory of analytic structures on manifolds.

The Cartan–Kähler theorem is a fundamental result in the field of differential geometry and partial differential equations, dealing with the integration of partial differential equations. It establishes conditions under which solutions exist for a certain class of systems of partial differential equations. Specifically, the theorem provides criteria for the existence of "integral submanifolds" of a given system of differential equations.

The Cauchy formula for repeated integration is a result in calculus that provides a way to express the \( n \)-th repeated integral of a function in terms of its derivatives. Specifically, it relates the \( n \)-fold integral of a function to its \( n \)-th derivative.

The Cauchy–Kovalevskaya theorem is a fundamental result in the theory of partial differential equations (PDEs) that provides conditions under which a certain class of initial value problems has solutions. Named after Augustin-Louis Cauchy and Sofia Kovalevskaya, the theorem essentially states that if the initial conditions of a certain type of PDE are satisfied, then there exists a unique analytic solution in a neighborhood of the initial value.

The Chebyshev–Markov–Stieltjes inequalities refer to a set of results in probability theory and analysis that provide estimates for the probabilities of deviations of random variables from their expected values. These inequalities are generalizations of the well-known Chebyshev inequality and are closely related to concepts from measure theory and Stieltjes integrals.

Danskin's theorem is a result in the field of optimization and convex analysis. It provides a result on the sensitivity of the optimal solution of a parametric optimization problem.

The Denjoy–Koksma inequality is a key result in the field of numerical integration and approximation theory, particularly in the context of uniform distribution theory. It provides a bound on the discrepancy of a sequence of points used in numerical integration and describes how well a given numerical method approximates the integral of a function.

The Denjoy–Young–Saks theorem is a result in measure theory concerning the decomposition of the Lebesgue measurable sets. It is named after mathematicians Arne Magnus Denjoy, John Willard Young, and Aleksandr Yakovlevich Saks, who contributed to the development of this area of mathematics.

The Dirichlet–Jordan test is a criterion used in analysis, particularly in the study of the convergence of series of functions, such as Fourier series. The test is useful for determining the pointwise convergence of a series of functions, especially when dealing with orthogonal functions or trigonometric series.

Fenchel's duality theorem is a fundamental result in convex analysis and optimization, which establishes a relationship between a convex optimization problem and its dual problem. Specifically, it provides conditions under which the solution of a primal convex optimization problem can be found by solving its dual.

The Fenchel–Moreau theorem is a fundamental result in convex analysis that relates the concepts of convex conjugates and duality. It characterizes the relationship between a convex function and its conjugate. Let \( f : \mathbb{R}^n \to \mathbb{R} \) be a proper, convex, and lower semicontinuous function.

The Fraňková–Helly selection theorem is a result in the field of functional analysis and topology, specifically concerning the selection of points from family of sets. It builds upon the classical Helly's theorem, which deals with finite intersections of convex sets in Euclidean spaces. The Fraňková–Helly selection theorem provides conditions under which one can extract a sequence from a family of sets that converges in a certain sense.

Fuchs' theorem is a result in the field of complex analysis, particularly in the study of ordinary differential equations with singularities. The theorem provides conditions under which a linear ordinary differential equation with an irregular singular point can be solved using power series methods. Specifically, Fuchs' theorem states that if a linear differential equation has only regular singular points, then around each regular singular point, there exist solutions that can be expressed as a Frobenius series.

The Gaussian integral refers to the integral of the function \( e^{-x^2} \) over the entire real line.

Glaeser's continuity theorem is a result in the field of real analysis, specifically concerning the continuity properties of certain functions. While I cannot provide the specific wording of the theorem, I can summarize its significance and implications. The theorem is often related to the concepts of continuity in functions defined on certain spaces. It typically deals with the conditions under which a function can be approximated continuously by other functions, or under which certain limits exist as parameters change.

Godunov's theorem is a result in the field of numerical analysis, specifically related to the numerical solution of hyperbolic partial differential equations (PDEs). It is named after the Russian mathematician S. K. Godunov, who contributed significantly to the development of finite volume methods for solving these types of equations.

The Goldbach–Euler theorem is a result in number theory that relates to the representation of even integers as sums of prime numbers. More specifically, it builds on the ideas of the original Goldbach conjecture. While the conjecture itself states that every even integer greater than 2 can be expressed as the sum of two prime numbers, the Goldbach–Euler theorem provides a more generalized framework.

Helly's selection theorem is a result in combinatorial geometry and convex analysis, named after the mathematician Eduard Helly. The theorem asserts conditions under which a family of convex sets possesses a point in common, based on the intersections of smaller subfamilies of those sets. The precise statement of Helly's selection theorem typically involves a finite collection of convex sets in \(\mathbb{R}^d\).