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42 relations: Analytic continuation, Branch point, Calculus, Cauchy distribution, Cauchy principal value, Cauchy's integral formula, Cauchy's integral theorem, Characteristic function (probability theory), Complex analysis, Complex number, Complex plane, Complex-valued function, Continuous function, Curve, Dirichlet series, Entire function, Equivalence class, Estimation lemma, Euler's identity, Functional equation, Hankel contour, Imaginary number, Integral, Interval (mathematics), Jordan curve theorem, Laurent series, Line integral, Numerical analysis, Parametrization, Partition of an interval, Pochhammer contour, Poisson kernel, Probability theory, Real line, Real number, Residue (complex analysis), Residue theorem, Riemann integral, Riemann zeta function, Singularity (mathematics), Special functions, Trigonometric functions.
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
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Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.
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Calculus
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
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Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.
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Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
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Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
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Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.
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Characteristic function (probability theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.
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Complex analysis
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.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Complex-valued function
In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
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Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.
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Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Estimation lemma
In mathematics the estimation lemma, also known as the inequality, gives an upper bound for a contour integral.
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Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler.
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Functional equation
In mathematics, a functional equation is any equation in which the unknown represents a function.
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Hankel contour
In mathematics, a Hankel contour is a path in the complex plane which extends from, around the origin counter clockwise and back to, where δ is an arbitrarily small positive number.
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Imaginary number
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit,j is usually used in Engineering contexts where i has other meanings (such as electrical current) which is defined by its property.
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
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Jordan curve theorem
In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting continuous loop in the plane.
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Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.
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Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.
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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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Parametrization
Parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.
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Partition of an interval
In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of.
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Pochhammer contour
In mathematics, the Pochhammer contour, introduced by and, is a contour in the complex plane with two points removed, used for contour integration.
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Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.
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Probability theory
Probability theory is the branch of mathematics concerned with probability.
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Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.
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Residue theorem
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 can often be used to compute real integrals as well.
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Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
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Singularity (mathematics)
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.
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Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.
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Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
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Calculus of residues, Complex integration, Contour integral, Examples of contour integrals, Examples of contour integration, Integration using complex analysis, Keyhole contour, Method of complex integration, Method of contour integration, Methods of contour integration, Residue calculus.
References
[1] https://en.wikipedia.org/wiki/Contour_integration
