Complex differential equation
Encyclopedia
A complex differential equation is a differential equation
whose solutions are functions of a complex variable.
Constructing integrals involves choice of what path to take, which means singularities and branch point
s of the equation need to be studied. Analytic continuation
is used to generate new solutions and this means topological considerations such as monodromy
, coverings
and connectedness
are to be taken into account.
Existence and uniqueness theorems involve the use of majorants and minorants.
Study of rational
second order ODE
s in the complex plane led to the discovery of new transcendental
special functions
which are now known as Painlevé transcendents
.
Nevanlinna theory
can be used to study complex differential equations. This leads to extensions of Malmquist's theorem.
s in several complex variables
, or differential equations on complex manifold
s. Also there are at least a couple of ways of studying complex difference equations: either study holomorphic functions which satisfy functional relations given by the difference equation or study discrete
analogs of holomorphicity such as monodiffric functions. Also integral equations can be studied in the complex domain.
Pierre Boutroux
, Paul Painlevé
, Lazarus Fuchs
, Henri Poincaré
, David Hilbert
, George David Birkhoff
, Kōsaku Yosida
, Hans Wittich, Charles Briot, Jean Claude Bouquet, Johannes Malmquist.
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
whose solutions are functions of a complex variable.
Constructing integrals involves choice of what path to take, which means singularities and branch point
Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...
s of the equation need to be studied. Analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
is used to generate new solutions and this means topological considerations such as monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
, coverings
Cover (topology)
In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
and connectedness
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
are to be taken into account.
Existence and uniqueness theorems involve the use of majorants and minorants.
Study of rational
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
second order ODE
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s in the complex plane led to the discovery of new transcendental
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
special functions
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....
which are now known as Painlevé transcendents
Painlevé transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property , but which are not generally solvable in terms of elementary functions...
.
Nevanlinna theory
Nevanlinna theory
Nevanlinna theory is a branch of complex analysis developed by Rolf Nevanlinna. It deals with the value distribution theory of holomorphic functions in one variable, usually denoted z....
can be used to study complex differential equations. This leads to extensions of Malmquist's theorem.
Generalizations
Generalizations include partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s in several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, or differential equations on complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s. Also there are at least a couple of ways of studying complex difference equations: either study holomorphic functions which satisfy functional relations given by the difference equation or study discrete
Discrete
Discrete in science is the opposite of continuous: something that is separate; distinct; individual.Discrete may refer to:*Discrete particle or quantum in physics, for example in quantum theory...
analogs of holomorphicity such as monodiffric functions. Also integral equations can be studied in the complex domain.
History
Some of the early contributors to the theory of complex differential equations include:Pierre Boutroux
Pierre Boutroux
Pierre Léon Boutroux was a French mathematician and historian of science. Boutroux is chiefly known for his work in the history and philosophy of mathematics.-Biography:...
, Paul Painlevé
Paul Painlevé
Paul Painlevé was a French mathematician and politician. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925.-Early life:Painlevé was born in Paris....
, Lazarus Fuchs
Lazarus Fuchs
Lazarus Immanuel Fuchs was a German mathematician who contributed important research in the field of linear differential equations...
, Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, George David Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
, Kōsaku Yosida
Kosaku Yosida
was a Japanese mathematician who worked in the field of functional analysis. He is known for the Hille-Yosida theorem concerning C0-semigroups.- References :...
, Hans Wittich, Charles Briot, Jean Claude Bouquet, Johannes Malmquist.
See also
- Frobenius methodFrobenius methodIn mathematics, the Frobenius method, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the formin the vicinity of the regular singular point z=0....
- Heun's equationHeun's equationIn mathematics, the local Heun function Hℓ is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0...
- Hypergeometric differential equationHypergeometric differential equationIn mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation...
- Riemann's differential equationRiemann's differential equationIn mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0,1, and ∞....
- Riemann–Hilbert problem
- Riemann–Hilbert correspondenceRiemann–Hilbert correspondenceIn mathematics, the Riemann–Hilbert correspondence is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups...
- Schwarzian derivativeSchwarzian derivativeIn mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and...
- Knizhnik–Zamolodchikov equations