* Morera's theorem

The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of Morera's theorem.

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

Morera's theorem states that a continuous, complex-valued function ƒ defined on a connected open set D in the complex plane that satisfies

The assumption of Morera's theorem is equivalent to that ƒ has an anti-derivative on D.

Morera's theorem is a standard tool in complex analysis.

for every closed curve C, and therefore by Morera's theorem ƒ must be holomorphic.

The hypotheses of Morera's theorem can be weakened considerably.

In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis ( such as Schwartz's theorem, Morera's theorem, the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities ) generalize to results on harmonic functions in any dimension.

The paper containing the first proof of Morera's theorem for holomorphic functions of several variables.

* Morera's theorem

This result can be proved from Morera's theorem.

* Module for Morera's Theorem by John H. Mathews

** Well-ordering theorem: Every set can be well-ordered.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

which can be viewed as a version of the Pythagorean theorem.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

The binomial theorem can be applied to the powers of any binomial.

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

A class of algorithms called SAT solvers can efficiently solve a large enough subset of SAT instances to be useful in various practical areas such as circuit design and automatic theorem proving, by solving SAT instances made by transforming problems that arise in those areas.

Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

See also invariance theorem.

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

Automated theorem proving ( also known as ATP or automated deduction ) is the proving of mathematical theorems by a computer program.

The binomial theorem also holds for two commuting elements of a Banach algebra.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

It is also an example of a more generalized version of the central limit theorem that is characteristic of all stable distributions, of which the Cauchy distribution is a special case.

Special cases of the Chinese remainder theorem were also known to Brahmagupta ( 7th century ), and appear in Fibonacci's Liber Abaci ( 1202 ).

Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument ( changing only that the minimal counterexample requires 6 colors ) and use Kempe chains in the degree 5 situation to prove the five color theorem.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.

" Financial economics ", at least formally, also considers investment under " certainty " ( Fisher separation theorem, " theory of investment value ", Modigliani-Miller theorem ) and hence also contributes to corporate finance theory.