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Encyclopedia > Hypergeometric differential equation

In quantum fitness, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. Every second-order linear ODE with three regular singular points can be transformed into this equation. The solutions are a special case of a Schwarz-Christoffel mapping to a triangle with circular arcs as edges. These are important because of the role they play in the theory of triangle groups, from which the inverse to Klein's J-invariant may be constructed. Thus, the solutions are coupled to the theory of Fuchsian groups and thus hyperbolic Riemann surfaces. A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In mathematics, in the theory of ordinary differential equations in the complex plane C, the points of C are classified into ordinary points, at which the equations coefficients are analytic functions, and singular points, at which some coefficient has a singularity. ... In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformally to a polygon. ... In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. ... Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

Contents

Definition

The hypergeometric differential equation is

z(1-z)frac {d^2w}{dz^2} + left[c-(a+b+1)z right] frac {dw}{dz} - abw = 0.

It has three regular singular points: 0,1 and ∞. The generalization of this equation to arbitrary regular singular points is given by Riemann's differential equation. In mathematics, Riemanns differential equation 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 ∞. // Definition The differential equation is given by The regular singular points are a, b and c. ...


Solutions

Solutions to the differential equation are built out of the hypergeometric series ;_2F_1(a,b;c;z). In general, the equation has two linearly independent solutions. One starts by defining the values In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

λ = 1 − c
μ = cab
ν = ab.

These are known as the angular parameters for the regular singular points 0,1 and ∞ respectively. Frequently, the notation ν0, ν1 and nu_infty, respectively, are used for the angular parameters. Sometimes, the exponents μ0, μ1, μz and mu_infty are used, with

mu_0=frac{1}{2}(1-nu_0+nu_1-nu_infty)= c-a
mu_1=frac{1}{2}(1+nu_0-nu_1-nu_infty)= b+1-c
mu_z=frac{1}{2}(1-nu_0-nu_1+nu_infty)= a
mu_infty=frac{1}{2}(1+nu_0+nu_1+nu_infty)= 1-b

and mu_0+mu_1+mu_z+mu_infty=2.


The general case, where none of the angular parameters are integers, is given below. When one or more of these parameters are integers, the solutions are given in the article hypergeometric equation solutions. The integers are commonly denoted by the above symbol. ...


Around the point z=0, the two independent solutions are

phi_0^{(0)}(z)= ;_2F_1(a,b;c;z)

and

phi_0^{(1)}(z) = z^lambda ;_2F_1(a+lambda,b+lambda;1+lambda;z)

Around z=1, one has

phi_1^{(0)}(z)= ;_2F_1(a,b;1-mu;1-z)

and

phi_1^{(1)}(z) = (1-z)^mu ;_2F_1(b+mu,a+mu;1+mu;1-z)

Around z=∞ one has

phi_infty^{(0)}(z) = z^{-a};_2F_1(a,a+lambda;1+nu; z^{-1})

and

phi_infty^{(1)}(z) = z^{-b};_2F_1(b,b+lambda;1-nu; z^{-1})

This is the complete set of solutions. Kummer's set of 24 canonical solutions may be obtained by applying either or both of the following identities to the above equations: Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ...

;_2F_1(a,b;c;z)= (1-z)^{c-a-b} ;_2F_1(c-a,c-b;c;z)

and

;_2F_1(a,b;c;z)=(1-z)^{-a} ;_2F_1(a,c-b;c;z/(z-1))

Connection coefficients

Pairs of solutions are related to each other through connection coefficients, corresponding to the analytic continuation of the solutions. Denote a pair of solutions as the column vector

Phi_k = left( begin{matrix} phi_k^{(0)}  phi_k^{(1)} end{matrix} right)

for k=0,1, ∞. Pairs are related by matrices

Phi_0 = left( begin{matrix} frac{Gamma(c)Gamma(c-a-b)}{Gamma(c-a)Gamma(c-b)} ; & ; frac{Gamma(c)Gamma(a+b-c)}{Gamma(a)Gamma(b)}  ; &;  frac{Gamma(2-c)Gamma(c-a-b)}{Gamma(1-a)Gamma(1-b)} ; & ; frac{Gamma(2-c)Gamma(a+b-c)}{Gamma(a+1-c)Gamma(b+1-c)} end{matrix} right) Phi_1

and

Phi_0 = left( begin{matrix} e^{-ipi a} frac{Gamma(c)Gamma(b-a)}{Gamma(c-a)Gamma(b)} ; & ; e^{-ipi b} frac{Gamma(c)Gamma(a-b)}{Gamma(c-b)Gamma(a)}  ; &;  e^{-ipi(a+1-c)} frac{Gamma(2-c)Gamma(b-a)}{Gamma(b+1-c)Gamma(1-a)} ; & ; e^{-ipi(b+1-c)} frac{Gamma(2-c)Gamma(a-b)}{Gamma(a+1-c)Gamma(1-b)} end{matrix} right) Phi_infty

where Γ is the gamma function. The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...


Q-form

The hypergeometric equation may be brought into the Q-form

frac{d^2u}{dz^2}+Q(z)u(z) = 0

by making the substitution w = uv and eliminating the first-derivative term. One finds that

Q=frac{z^2[1-(a-b)^2] +z[2c(a+b-1)-4ab] +c(2-c)}{4z^2(1-z)^2}

and v is given by the solution to

frac{d}{dz}log v(z) = frac {c-z(a+b+1)}{2z(1-z)}

The Q-form is significant in its relation to the Schwarzian derivative. In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ...


Schwarz triangle maps

The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.

s_k(z) = frac{phi_k^{(1)}(z)}{phi_k^{(0)}(z)}

where k is one of the points 0,1, ∞. The notation

Dk(λ,μ,ν;z) = sk(z)

is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps. In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...


Note that each triangle map is regular at z=0,1 and ∞ respectively, with In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ...

s_0(z)=z^lambda (1+mathcal{O}(z))
s_1(z)=(1-z)^mu (1+mathcal{O}(1-z))

and

s_infty(z)=z^nu (1+mathcal{O}(1/z))

In the special case of λ, μ and ν real, with 0le|lambda|,|mu|,|nu|<1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a special case of a Schwarz-Christoffel map. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively. In mathematics, a conformal map is a function which preserves angles. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... A rendering of the Riemann Sphere. ... In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformally to a polygon. ...


Furthermore, in the case of λ = 1 / p, μ = 1 / q and ν = 1 / r for integers p, q, r, then the triangle tiles the sphere, and the s-maps are inverse functions of automorphic functions for the triangle group langle p,q,rrangle=Delta (p,q,r). In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. ...


Monodromy group

The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point. That is, when the path winds around a singularity of ;_2F_1, the value of the solutions at the endpoint will differ from the starting point.


Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):

pi_1(z_0,mathbb{C}setminus{0,1}) to GL(2,mathbb{C})

where π1 is the fundamental group. In other words the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ...


See also

In mathematics, the Heuns differential equation is a second-order linear ordinary differential equation (ODE) of the form (Note that is needed to ensure regularity of the point at ∞.) Every second-order linear ODE in the complex plane (or on the Riemann sphere, to be more accurate) with four... In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. ... In mathematics, a Schottky group is a special sort of Kleinian group, named after Friedrich Schottky. ... In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ...

References

  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
    • Chapter 15 Hypergeometric Functions
      • Section 15.5 The Hypergeometric Differential Equation
  • Frits Beukers, Gauss' hypergeometric function (2002) (Lecture notes reviewing basics, as well as triangle maps and monodromy)
  • Masaaki Yoshida (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Friedrick Vieweg & Son. ISBN 3-528-06925-2. 

  Results from FactBites:
 
Hypergeometric series - Wikipedia, the free encyclopedia (1015 words)
In mathematics, a hypergeometric series is the sum of a sequence of terms in which the ratios of successive coefficients k is a rational function of k.
Applications of hypergeometric series includes the inversion of elliptic integrals; these are constructed by taking the ratio of the two linearly independent solutions of the hypergeometric differential equation to form Schwarz-Christoffel maps of the fundamental domain to the complex projective line or Riemann sphere.
Riemann showed that the second-order differential equation (in z) for F, examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities: that effectively the entire algorithmic side of the theory was a consequence of basic facts and the use of Möbius transformations as a symmetry group.
  More results at FactBites »

 
 

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