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Lindemann–Weierstrass Theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transcendence degree over \mathbb. An equivalent formulation from , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over \mathbb by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transcendental Number Theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendence The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial P with rational coefficients there will be a complex number \alpha such that P(\alpha)=0. Transcendence theory is concerned with the converse question: given a complex number \alpha, is there a polynomial P with rational coefficients such that P(\alpha)=0? If no such polynomial exists then the number is called transcendental. More generally the theory deals with algebraic independence of numbers. A set of numbers is called algebraically independen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Integer
An integer is the number zero ( 0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lemma (mathematics)
In mathematics and other fields, a lemma (: lemmas or lemmata) is a generally minor, proven Theorem#Terminology, proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to mathematical proof, prove; however, a lemma can also turn out to be more important than originally thought. Etymology From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument. Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem#Terminology, Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas Some powerful results in mathematics are known as lemmas, first named for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Unit Disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees. Definition The nome function is given by :q =\mathrm^ =\mathrm^ =\mathrm^ \, where K and iK' are the quarter periods, and \omega_1 and \omega_2 are the fundamental pair of periods, and \tau=\frac=\frac is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when 0 |
J-invariant
In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that :j\big(e^\big) = 0, \quad j(i) = 1728 = 12^3. Rational functions of j are modular, and in fact give all modular functions of weight 0. Classically, the j-invariant was studied as a parameterization of elliptic curves over \mathbb, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane \mathcal=\, by :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a cube, also since 1728 = 12^3. The function cannot be continued analytically beyond the upper half-plane due to the natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modular Function
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-adic Exponential Function
In mathematics, particularly ''p''-adic analysis, the ''p''-adic exponential function is a ''p''-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the ''p''-adic logarithm. Definition The usual exponential function on C is defined by the infinite series :\exp(z)=\sum_^\infty \frac. Entirely analogously, one defines the exponential function on C''p'', the completion of the algebraic closure of Q''p'', by :\exp_p(z)=\sum_^\infty\frac. However, unlike exp which converges on all of C, exp''p'' only converges on the disc :, z, _p [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-adic Numbers
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hyperbolic Function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Euler's Identity
In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definition satisfies i^2 = -1, and :\pi is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula e^ = \cos x + i\sin x when evaluated for x = \pi. Euler's identity is considered an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in Lindemann–Weierstrass theorem#Transcendence of e and π, a proof that is Transcendental number, transcendental, which implies the impossibility of squaring the circle. Mathematical beauty Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
