pole

From Middle French pole, pôle, from Latin polus, from Ancient Greek πόλος (pólos, “axis of rotation”).

der

The function f(z)#61;#92;frac#123;1#125;#123;z-3#125; has a single pole at z#61;3.

(complex analysis) For a meromorphic function f(z), any point a for which f(z)→∞ as z→a.

pól

isolated point of a meromorphic function (complex analysis)


entire

From Middle English entere, enter, borrowed from Anglo-Norman entier, from Latin integrum, accusative of integer (“whole”), from Proto-Italic *əntagros (“untouched”). Doublet of entier and integer.

(complex analysis, of a complex function) Complex-differentiable on all of ℂ.

analyyttinen

complex analysis: differentiable everywhere


complex number

LL-Q1860 (eng)-Flame, not lame-complex number.wav

(complex analysis) A number of the form a + bi, where a and b are real numbers and i denotes the imaginary unit.


imaginary number

en-noun

imaginary number (plural imaginary numbers)

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Carl Friedrich Gauss
Leonhard Euler
René Descartes
imaginary number

The adjective imaginary in this context was first used (as French imaginaire) by René Descartes in 1673, La Geometrie, referring to imaginary numbers in the broad sense, as non-real roots of polynomials. Descartes' usage was derogatory, but the concept later gained acceptance through the work of Leonhard Euler and Carl Friedrich Gauss in the 18th century.

If a#61;0, then x#61;ib is called an imaginary number. The message here is that we must introduce imaginary numbers in order to be able to solve quadratic equations in general.

2008, Donald A. McQuarrie, Mathematics for Physical Chemistry: Opening Doors, University Science Books, page 54:

(complex analysis, strict sense) A number of the form bi, where b is any real number and i denotes the imaginary unit.

(complex analysis, broad sense) A number of the form a + bi, where a and b are real numbers and b is nonzero.


imaginary part

(complex analysis) Of a complex number a + bi, where a and b are real numbers, the value b.


real part

(complex analysis) Of a complex number a + bi, the value a.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(complex analysis, of a family of continuous functions) Which is pre-compact.


univalent

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(complex analysis, of a (holomorphic) function) (Holomorphic and) injective.


transcendental function

en-noun

transcendental function (plural transcendental functions)

(specifically, complex analysis) A meromorphic function on the complex plane that is not a rational function.


residue

residue (countable and uncountable, plural residues)

From Middle English residue, from Old French residu, from Latin residuum, neuter of residuus (“remaining”), from resideō (“I remain behind”).

By taking residues, we can represent this integral in terms of a Jackson integral.

2001, Publications of the Research Institute for Mathematical Sciences, page 72:

(complex analysis) A form of complex number, proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.


purely imaginary number

purely imaginary number (plural purely imaginary numbers)


purely imaginary

(complex analysis) Of a number, having real part equal to zero.


rational function

rational function (plural rational functions)

Our first problem is that of interpolation in prescribed points to a given function by a rational function whose poles are given.

1960, J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 3rd edition, American Mathematical Society, page 184:

By Theorem 2.3.2., we have that the right-hand side of this equation can be equal to a rational function only if that rational function is equal to zero.

1970, Ellis Horowitz, Algorithms for Symbolic Integration of Rational Functions, University of Wisconsin-Madison, page 24:

Let #92;mathcal#123;C#125; be the class of continuous maps of #92;mathbb#123;C#125;#95;#92;infty into itself and let #92;mathcal#123;R#125; be the subclass of rational functions.[…]Now #92;mathcal#123;R#125; is a closed subset of #92;mathcal#123;C#125;#95;#92;infty because if the rational functions R#95;n converge uniformly to R on the complex sphere, then R is analytic on the sphere and so it too is rational.

2000, Alan F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, Springer, page 45:

(mathematics, complex analysis, algebraic geometry) Any function expressible as the quotient of two (coprime) polynomials (and which thus has poles at a finite, discrete set of points which are the roots of the denominator).

rationaalifunktio

function expressible as the quotient of polynomials


Euler's identity

Named after Swiss mathematician Leonhard Euler (1707–1783).

named-after

(complex analysis) The equation e^(iπ)=-1,!, which unifies diverse fields of mathematics.


Euler's formula

eⁱᶿ= cos θ+i sin θ

trigonometric function
trigonometric function

(complex analysis) A formula which links complex exponentiation with trigonometric functions:


meromorphic

Normal families of meromorphic functions are most naturally studied using the spherical metric (§1.2), an approach initiated by Ostrowski [1926]. Some results for meromorphic functions, such as the FNT, are immediate extensions from the analytic case, whereas others, such as Landau's or Julia's theorem are set in a much broader context than their analytic counterparts. Normality criteria pertinent to families of meromorphic functions, such as Marty's theorem, have not yet been encountered.

1993, Joel L. Schiff, Normal Families, Springer, page 71:

Note that such a transformation is holomorphic at the origin, but is essentially singular at infinity. However, since T(z) commutes with A(z), the transformed system has coefficient matrix A(z)-zq'(z)I and hence is again meromorphic at infinity.

2000, Werner Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, page 39:

A point p#92;inP¹ is singular for #92;textstyle#92;frac#123;d#125;#123;dz#125;-A if the equation cannot be made regular at p with a local meromorphic transformation.

2012, Marius van der Put, Michael F. Singer, Galois Theory of Linear Differential Equations, Springer, page 147:

(complex analysis, of a function) That is the ratio of two holomorphic functions (and so possibly infinite at a discrete set of points).


holomorphic

1998, Stefan-M. Heinemann, Julia Sets in Cⁿ, Hartje Kriete, Progress in Holomorphic Dynamics, Addison Wesley Longman, page 159,
The iteration of holomorphic maps of one variable is one of the most lively fields of current mathematical research. Since its beginning in the twenties of this century, the theory of holomorphic endomorphisms of the Riemann sphere (= rational functions) has become well developed.

1998, Robert Friedman, Algebraic Surfaces and Holomorphic Vector Bundles, Springer, page 303,
A holomorphic 1-form on X is closed, and so
0=dη₁=df and η₂.

The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector field) acting on it.

2010, Graziano Gentili, Jacques Guenot, Giorgio Patrizio, Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School, Springer, page v:

(complex analysis, of a complex function) Complex-differentiable on an open set around every point in its domain.


bilinear

bilinear isotropic hardening

Möbius transformation
Möbius transformation

(complex analysis, physics, engineering) Of or pertaining to a Möbius transformation (type of conformal map representable as the ratio of two linear functions).


imaginary unit

So named because it takes on the role of unit for the imaginary part of a complex number.

In Section 4.3 we introduced the imaginary unit i with (i)²#61;-1, in the complex space domain. In section 4.4 we introduced the complex unit j with (j)²#61;-1, in connection with the time dependency of the shape (4.4.16) of the swimming profile. These different imaginary units have no interaction, we leave unaltered the product ij.

2013, J. A. Sparenberg, Hydrodynamic Propulsion and Its Optimization, Springer, page 183:

2014, Dennis G. Zill, Warren S. Wright, Advanced Engineering Mathematics, Ascend Learning (Jones & Bartlett Learning), 5th Edition, page 793,
We now simply say that i is the imaginary unit and define it by the property i²=-1. Using the imaginary unit, we build a general complex number out of two real numbers.

Consider now in addition to the given #92;H#92;subset#92;mathbb#123;O#125;, one of the totally imaginary units not contained in #92;H, and note that #123;SU#125;(2) acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming #92;alpha#92;notin#92;H, we may in fact assume that #92;mathbb#123;O#125;#61;#92;H#92;oplus#92;H' with #92;alpha#92;in#92;H'.

2021, Bruce Hunt, Locally Mixed Symmetric Spaces, Springer, page 123:

(number theory, complex analysis, quaternion theory) An imaginary number (in the case of complex numbers, usually denoted i) that is defined as a solution to the equation x²=-1.


Riemann sphere

Named after German mathematician Bernhard Riemann.

We use #92;hat#123;#92;Complex#125;#92;times (resp. #92;Complex) to denote the subset #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne 0#92;#125; (resp. #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne#92;infty#92;#125;) in the Riemann sphere #92;hat#92;Complex.

2002, Wenhua Zhao, “Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory”, in Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, Brian Parshall, editors, Recent Developments in Infinite-dimensional Lie Algebras and Conformal Field Theory, American Mathematical Society, page 306:

(topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.

1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,
Every circle γ on the Riemann sphere Σ which does not go through a given point P^*∈Σ divides Σ into two parts, such that one part contains P^* and the other does not.

In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#47;2 and is tangent to the complex plane at the origin (see Figure 1.8). We call the point of contact the south pole (denoted by S) and the point diametrically opposite S the north pole (denoted by N). Let z be an arbitrary point in the complex plane, represented by the point P. We draw the line PN which intersects the Riemann sphere at the unique point P', distinct from N. Conversely, to each point P' on the sphere, other than the north pole N, we draw the line P'N which cuts uniquely one point P in the complex plane. Clearly, there exists a one-to-one correspondence between points on the Riemann sphere, except N, and all the finite points on the complex plane. We assign the north pole N as the point at infinity. With such an assignment, we then establish a one-to-one correspondence between all the points on the Riemann sphere and all the points in the extended complex plane. This correspondence is known as the stereographic projection.

2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27:

We find that many mathematicians, even those who specialize in complex analysis and conformal geometry, are not familiar with the inversive distance between pairs of circles in the Riemann sphere.

2003, Philip L. Bowers, Monica K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces”, in Hans-Christian Hege, Konrad Polthier, editors, Visualization and Mathematics III, Springer, page 6:

extended complex plane
extended complex plane

extended complex numbers
extended complex numbers

(topology, complex analysis) The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.

complex projective line

differently constructed set homeomorphic to the Riemann sphere


complex-differentiable

This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany^([sic]) deep results.

1993, Victor Khatskevich, David Shoiykhet, Differentiable Operators and Nonlinear Equations, page 75:

Further it can be shown that the holomorphic function also has a convergent Taylor series, that is, a complex differentiable function is also an analytic function (Section 23.7).

2010, Luis Manuel Braga da Costa Campos, Complex Analysis with Applications to Flows and Fields, page 335:

We can of course regard a function f defined on #92;mathbb#123;R#125;#123;2n#125; as a function defined on #92;mathbb#123;C#125;#123;n#125;. If f is differentiable on #92;mathbb#123;R#125;#123;2n#125;, it is said to be real-differentiable, and if f is differentiable on #92;mathbb#123;C#125;#123;n#125;, it is complex-differentiable. A function is complex-differentiable if and only if it is real-differentiable and the Cauchy-Riemann equations hold.

2010, Peter J. Schreier, Louis L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, page 277:

Cauchy-Riemann equation
Cauchy-Riemann equation

(mathematics, complex analysis, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane.

analytic

differentiable and that satisfies the Cauchy-Riemann Equations on a subset of the complex plane

entire

differentiable and that satisfies the Cauchy-Riemann Equations on the complex plane


Möbius transformation

en-noun

Möbius transformation (plural Möbius transformations)

Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).

2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,
Jørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete.

2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,
Modular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant.

Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.

2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1:

complex projective line
complex projective line

(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.

August Ferdinand Möbius
Möbius transformation


branch cut

(topology, complex analysis) A curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve.


Cauchy-Riemann equation

Cauchy-Riemann equation (plural Cauchy-Riemann equations)

Augustin Cauchy
Bernhard Riemann
Cauchy-Riemann equation

Named after mathematicians Augustin Cauchy (1789-1857) and Bernhard Riemann (1826-1866).

(mathematics, complex analysis, always plural) Given a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable.

(complex analysis) The equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0.


stereographic projection

stereographic projection (plural stereographic projections)

1974 [Prentice-Hall], Richard A. Silverman, Complex Analysis with Applications, Dover, 1984, Unaltered republication, page 19,
Thus we see that stereographic projection establishes a one-to-one correspondence between the extended complex plane and the Riemann sphere.

1991, E. J. F. Primrose (translator), Simeon Ivanov (editor of translation), A. T. Fomenko, A. A. Tuzhilin, Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space, American Mathematical Society, page 67,
Proposition 2. The stereographic projection π_N:S²→Oxy of the sphere S² onto the xy-plane from the North Pole N preserves the angles between tangent vectors. Thus, the coordinates specified on S²⧵N by the stereographic projection π_N are isothermal.

Before we can recognize such patterns confidently, we need to know whether the stereographic projection faithfully represents the true clustering of directions in space.

2004, Richard J. Lisle, Peter R. Leyshon, Stereographic Projection Techniques for Geologists and Civil Engineers, 2nd edition, Cambridge University Press, page 40:

(projective geometry, complex analysis, cartography) A function that maps a sphere onto a plane; especially, the map generated by projecting each point of the sphere from the sphere's (designated) north pole to a point on the plane tangent to the south pole.


complex line

1990, R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Volume 1 Function Theory, Wadsworth & Brooks/Cole, page 102,
However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in C ⁿ through a point A∈ C ⁿ in the direction of a vector B∈ C ⁿ is the one-dimensional complex submanifold of C ⁿ described parametrically as A+tB:t∈ C.

A complex line in #92;Cⁿ is a set of the form l#61;#92;#123;z#58;z#61;a#43;#92;lambdab,#92;lambda#92;in#92;C#92;#125;, where a and b are fixed points in #92;Cⁿ, with b#92;ne 0. Let us say that l is the complex line through a in the “direction” b.

2014, Paul M. Gauthier, Lectures on Several Complex Variables, Springer (Birkhäuser), page 39:

The question of finding different familes^([sic]) of complex lines sufficient for holomorphic extension was put in [12]. Clearly, the family of complex lines passing through one point is not enough. As shown in [16], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient.

2018, Bairambay Otemuratov, “A Mulitidimensional Analogue of Hartogs's Theorem on n-Circular Domains for Integrable Functions”, in Zair Ibragimov, Norman Levenberg, Utkir Rozikov, Azimbay Sadullaev, editors, Algebra, Complex Analysis, and Pluripotential Theory: 2 USUZCAMP, 2017, Springer,, page 110:

(complex analysis, analytic geometry) A 1-dimensional affine subspace of a vector space over the complex numbers.

complex plane

affine subspace that is 1-dimensional over the complex numbers


Nevanlinna theory

Named after Finnish mathematician Rolf Nevanlinna (1895–1980), who published the theory in 1925.

A key tool in Nevanlinna theory is the Nevanlinna characteristic, T(r,f), which measures the rate of growth of a meromorphic function.

Precisely, our aim has been to show how the Nevanlinna theory may be applied to get insight into the properties of solutions of complex differential equations.

1992, Ilpo Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, page 1:

Motivated by an analogy between Nevanlinna theory and Diophantine approximation theory, discovered independently by C. F. Osgood [Osg 1985] and P. Vojta [Vojt 1987], S. Lang recognized that the careful study of the error term in Nevanlinna'a Second Main Theorem would be of interest in itself.

2001, William Cherry, Zhuan Ye, Nevanlinna's Theory of Value Distribution, Springer, page vi:

2010, Paul Vojta, Diophantine Approximation and Nevanlinna theory, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta (editors), Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School, Springer, Lecture Notes in Mathematics 2009, page 111,
Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth's theorem on diophantine approximation. […] The circle of ideas has developed further in the last 20 years: Lang's conjecture on sharpening the error term in Roth's was carried over to a conjecture in Nevanlinna theory which was proved in many cases.

(complex analysis) A part of the theory of meromorphic functions that describes the asymptotic distribution of solutions to the equation ƒ(z) = a, as a varies.

Acta Mathematica
Nevanlinna theory
Rolf Nevanlinna


Ahlfors theory

Named after Finnish mathematician Lars Ahlfors (1907—1996), who published the theory in 1935.

In this chapter we use Ahlfors' theory of covering surfaces to obtain results on the functional n(r,a).

1968, Joseph Belsley Miles, The Asymptotic Behavior of the Counting Function for the A-values of a Meromorphic Function, University of Wisconsin-Madison, page 29:

Terms of the form o(A(r)) in Ahlfors theory are given in the form cD(r) where c is a constant.

1986, Pacific Journal of Mathematics, volumes 122-123, page 441:

The Ahlfors theory itself describes covering of curves or domains, but not covering of distinct, complex values #92;boldsymbol#123;a#125;.

2004, G. Barsegian, “A new program of investigations in analysis: Gamma-lines approaches”, in G. Barsegian, I. Laine, C. C. Yang, editors, Value Distribution Theory and Related Topics, Kluwer Academic, page 43:

differential geometry
differential geometry

bordered Riemann surface
bordered Riemann surface

(complex analysis, differential geometry) A geometric counterpart to Nevanlinna theory that extends the applicability of the concept of covering surface (of a topological space) by defining a covering number (a generalised "degree of covering") applicable to any bordered Riemann surface equipped with a conformal Riemannian metric.

Acta Mathematica
Ahlfors theory
Lars Ahlfors


Mittag-Leffler star

Mittag-Leffler star (plural Mittag-Leffler stars)

(complex analysis) A set in the complex plane obtained by attempting to extend a given complex-analytic function along rays emanating from a given point.

Gösta Mittag-Leffler
Mittag-Leffler star


Mittag-Leffler function

The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian function.

(complex analysis) A complex function defined as E_α,β(z)=∑ₖ₌₀ ᪲(zᵏ)/(Γ(αk+β)), where Γ(x) is the gamma function.

Gösta Mittag-Leffler
Mittag-Leffler function


Mittag-Leffler summation

Mittag-Leffler summation (countable and uncountable, plural Mittag-Leffler summations)

(complex analysis) Any of several variations of the Borel summation method for summing possibly divergent formal power series.

Gösta Mittag-Leffler
Mittag-Leffler summation


Schwarz-Christoffel mapping

Schwarz-Christoffel mapping (plural Schwarz-Christoffel mappings)

They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869.

(complex analysis, potential theory) A conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon; used in minimal surfaces, hyperbolic art, and fluid dynamics.


Gauss-Lucas theorem

Named after Carl Friedrich Gauss and Félix Lucas.

(complex analysis) A theorem that gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. It states that the roots of P′ all lie within the convex hull of the roots of P.


complex conjugate root theorem

(mathematics, complex analysis) The theorem that if a complex number is a root for a polynomial with real coefficients, the complex conjugate of that number will also be a root.