Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Complex analysisComplex 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 helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Liouville's theorem (complex analysis)In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all is constant. Equivalently, non-constant holomorphic functions on have unbounded images. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant.
Argument (complex analysis)In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in Figure 1. It is a multivalued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval .
Cauchy productIn mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
Laurent seriesIn mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
Augustin-Louis CauchyBaron Augustin-Louis Cauchy (UKˈkoʊʃi,_ˈkaʊʃi , USkoʊˈʃiː , oɡystɛ̃ lwi koʃi; 21 August 1789 - 23 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He (nearly) single-handedly founded complex analysis and the study of permutation groups in abstract algebra.
Series (mathematics)In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
Cauchy's integral formulaIn mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.
Cauchy condensation testIn mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence of non-negative real numbers, the series converges if and only if the "condensed" series converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original. The Cauchy condensation test follows from the stronger estimate, which should be understood as an inequality of extended real numbers.
GC-contentIn molecular biology and genetics, GC-content (or guanine-cytosine content) is the percentage of nitrogenous bases in a DNA or RNA molecule that are either guanine (G) or cytosine (C). This measure indicates the proportion of G and C bases out of an implied four total bases, also including adenine and thymine in DNA and adenine and uracil in RNA. GC-content may be given for a certain fragment of DNA or RNA or for an entire genome.
Real analysisIn mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.
Cauchy sequenceIn mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.
Complex numberIn mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the , and b is called the . The set of complex numbers is denoted by either of the symbols or C.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Mathematical economicsMathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Perovskite solar cellA perovskite solar cell (PSC) is a type of solar cell that includes a perovskite-structured compound, most commonly a hybrid organic–inorganic lead or tin halide-based material as the light-harvesting active layer. Perovskite materials, such as methylammonium lead halides and all-inorganic cesium lead halide, are cheap to produce and simple to manufacture. Solar-cell efficiencies of laboratory-scale devices using these materials have increased from 3.8% in 2009 to 25.
MathematicsMathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.
Formal power seriesIn mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form where is the th power of a variable ( is a non-negative integer), and is called the coefficient.
Mathematical modelA mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science).
