In mathematics, an exponential function is a function of the form where b is a positive real number, in which the argument x occurs as an exponent. For real numbers c and d, a function of the form f = a b c x + d is an exponential function, as it can be rewritten as a b c x + d = x; as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: For b = 1 the real exponential function is a constant and the derivative is zero because log e b = 0, for positive a and b > 1 the real exponential functions are monotonically increasing, because the derivative is greater than zero for all arguments, for b < 1 they are monotonically decreasing, because the derivative is less than zero for all arguments. The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself: Since changing the base of the exponential function results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or "the exponential function" and denoted by While both notations are common, the former notation is used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.
The exponential function satisfies the fundamental multiplicative identity This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation f = f f is an exponential function, f: R → R, x ↦ b x, with b > 0. The fundamental multiplicative identity, along with the definition of the number e as e1, shows that e n = e × ⋯ × e ⏟ n terms for positive integers n and relates the exponential function to the elementary notion of exponentiation; the argument of the exponential function can be any real or complex number or an different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable; this occurs in the natural and social sciences.
The graph of y = e x is upward-sloping, increases faster as x increases. The graph always lies above the x-axis but can be arbitrarily close to it for negative x; the slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function. Its inverse function is the natural logarithm, denoted log, ln, or log e; the real exponential function exp: R → R can be characterized in a variety of equivalent ways. Most it is defined by the following power series: exp := ∑ k = 0 ∞ x k k! = 1 + x + x 2 2 + x 3 6 + x 4 24 + ⋯ Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers z ∈ C (see below for the extension
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. By extension, use of complex analysis has applications in engineering fields such as nuclear, aerospace and electrical engineering; as a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is concerned with analytic functions of a complex variable. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Riemann, Cauchy and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is used throughout analytic number theory. In modern times, it has become popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.
Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are supposed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values z from the domain and their images f in the range may be separated into real and imaginary parts: z = x + i y and f = f = u + i v, where x, y, u, v are all real-valued. In other words, a complex function f: C → C may be decomposed into u: R 2 → R and v: R 2 → R, i.e. into two real-valued functions of two real variables. Any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: or, alternatively, as a vector-valued function from X into R 2; some properties of complex-valued functions are nothing more than the corresponding properties of vector valued functions of two real variables.
Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have different properties. In particular, every differentiable complex function is analytic, two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain; the latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, trigonometric functions. Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω. In the context of complex analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C.
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we
Karl Theodor Wilhelm Weierstrass was a German mathematician cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher teaching mathematics, physics and gymnastics. Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born in part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, Theodora Vonderforst, his interest in mathematics began. He was sent to the University of Bonn upon graduation to prepare for a government position; because his studies were to be in the fields of law and finance, he was in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics.
The outcome was to leave the university without a degree. After that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster, he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he taught physics and gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction; the University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin.
At the age of fifty-five, Weierstrass met Sonya Kovalevsky whom he tutored after failing to secure her admission to the University. They had a fruitful intellectual, but troubled personal relationship that "far transcended the usual teacher-student relationship"; the misinterpretation of this relationship and Kovalevsky's early death in 1891 was said to have contributed to Weierstrass' ill-health. He was immobile for the last three years of his life, died in Berlin from pneumonia. Weierstrass was interested in the soundness of calculus, at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 his work remained unknown to most of the mathematical community until years and many mathematicians had only vague definitions of limits and continuity of functions. Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.
Cauchy did not distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars; the correct statement is. This required the concept of uniform convergence, first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, both formalized it and applied it throughout the foundations of calculus; the formal definition of continuity of a function, as formulated by Weierstrass, is as follows: f is continuous at x = x 0 if ∀ ε > 0 ∃ δ > 0 such that for every x in the domain of f, | x − x 0 | < δ ⇒ | f − f | < ε. In simple English, f is continuous at a point x = x 0 if for each ε > 0 there exists a δ > 0 such that the function f lies between f − ε and f
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category essential singularity is a "left-over" or default group of isolated singularities that are unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. Consider an open subset U of the complex plane C. Let a be an element of U, f: U ∖ → C a holomorphic function; the point a is called an essential singularity of the function f if the singularity is neither a pole nor a removable singularity. For example, the function f = e 1 / z has an essential singularity at z = 0. Let a be a complex number, assume that f is not defined at a but is analytic in some region U of the complex plane, that every open neighbourhood of a has non-empty intersection with U. If both lim z → a f and lim z → a 1 f exist a is a removable singularity of both f and 1/f. If lim z → a f exists but lim z → a 1 f does not exist a is a zero of f and a pole of 1/f.
If lim z → a f does not exist but lim z → a 1 f exists a is a pole of f and a zero of 1/f. If neither lim z → a f nor lim z → a 1 f exists a is an essential singularity of both f and 1/f. Another way to characterize an essential singularity is that the Laurent series of f at the point a has infinitely many negative degree terms. A related definition is that if there is a point a for which no derivative of f n converges to a limit as z tends to a a is an essential singularity of f; the behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity a, the function f takes on every complex value, except one, infinitely many times. An Essential Singularity by Stephen Wolfram, Wolfram Demonstrations Project. Essential Singularity on Planet Math
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL defined on the upper half-plane of complex numbers. It is the unique such function, holomorphic away from a simple pole at the cusp such that j = 0, j = 1728 = 12 3. Rational functions of j are modular, in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it has surprising connections to the symmetries of the Monster group. While the j-invariant can be defined purely in terms of certain infinite sums, these can be motivated by considering isomorphism classes of elliptic curves; every elliptic curve E over C is a complex torus, thus can be identified with a rank 2 lattice. This is done by identifying opposite edges of each parallelogram in the lattice. However, multiplying the lattice by a complex number, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, so we can always arrange for the lattice to be generated by 1 and some τ in H. Conversely, if we define g 2 = 60 ∑ ≠ − 4, g 3 = 140 ∑ ≠ − 6 this lattice corresponds to the elliptic curve over C defined by y2 = 4x3 − g2x − g3 via the Weierstrass elliptic functions.
The j-invariant is defined as j = 1728 g 2 3 Δ where the modular discriminant Δ is Δ = g 2 3 − 27 g 3 2 It can be shown that Δ is a modular form of weight twelve, g2 one of weight four, so that its third power is of weight twelve. Thus their quotient, therefore j, is a modular function of weight zero, in particular a holomorphic function H → C invariant under the action of SL; as explained below, j is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over C and the complex numbers. The two transformations τ → τ + 1 and τ → -τ−1 together generate the special linear group SL. Quotienting out by its centre yields the modular group, which we may identify with the projective special linear group PSL. By a suitable choice of transformation belonging to this group, τ ↦ a τ + b c τ + d, a d − b c = 1, we may reduce τ to a value giving the same value for j, lying in the fundamental region for j, which consists of values for τ satisfying the conditions | τ | ≥ 1 − 1 2 < R ≤ 1 2 − 1 2 < R < 0 ⇒ | τ | > 1 The function j when restricted to this region still takes on every value in the complex numbers C once.
In other words, for every c in C, there is a unique τ in the fundamental region such that c = j. Thus, j has the property of mapping the fundamental region to the entire complex plane; as a Riemann surface, the fundamental region has genus 0, every modular function is a rational function in j. In other words, the field of modular functions is C. T
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves. It generalizes Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem; the statement is as follows: Let U be a connected open subset of the complex plane containing a finite list of points a1... an, f a function defined and holomorphic on U \. Let γ be a closed rectifiable curve in U which does not meet any of the ak, denote the winding number of γ around ak by I; the line integral of f around γ is equal to 2πi times the sum of residues of f at the points, each counted as many times as γ winds around the point: ∮ γ f d z = 2 π i ∑ k = 1 n I Res . If γ is a positively oriented simple closed curve, I = 1 if ak is in the interior of γ, 0 if not, so ∮ γ f d z = 2 π i ∑ Res with the sum over those ak inside γ; the relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem.
The general plane curve γ must first be reduced to a set of simple closed curves whose total is equivalent to γ for integration purposes. The requirement that f be holomorphic on U0 = U \ is equivalent to the statement that the exterior derivative d = 0 on U0, thus if two planar regions V and W of U enclose the same subset of, the regions V \ W and W \ V lie in U0, hence ∫ V ∖ W d − ∫ W ∖ V d is well-defined and equal to zero. The contour integral of f dz along γj = ∂V is equal to the sum of a set of integrals along paths λj, each enclosing an arbitrarily small region around a single aj — the residues of f at. Summing over, we recover the final expression of the contour integral in terms of the winding numbers. In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed, a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle.
The integral over this curve can be computed using the residue theorem. The half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were interested in; the integral ∫ − ∞ ∞ e i t x x 2 + 1 d x arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals. Suppose t > 0 and define the contour C that goes along the real line from −a to a and counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. Now consider the contour integral ∫ C f d z = ∫ C e i t z z 2 + 1 d z. Since eitz is an entire function, this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 =, that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour.
Because f is e i t z z 2 + 1 = e i t z 2 i ( 1 z − i − 1 z +