Schwarz_triangle

Schwarz triangle map

In mathematics, 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.

1 Definition
2 Solutions
3 Connection coefficients
4 Q-form
5 Schwarz triangle maps
6 Monodromy group
7 See also
8 References

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.

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

λ = 1 - c

μ = c - a - b

ν = a - b .

These are known as the angular parameters for the regular singular points 0,1 and ∞ respectively. Frequently, the notation $ν 0$ , $ν 1$ and $\nu_\infty$ , respectively, are used for the angular parameters. Sometimes, the exponents $μ 0$ , $μ 1$ , $μ z$ 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.

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:

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

and

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

For a solution of this differential equation using Frobenius method, please check Frobenius solution to the hypergeometric equation.

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

$\Phi_0 = \left( \begin{matrix} e^{-i\pi a} \frac{\Gamma(c)\Gamma(b-a)}{\Gamma(c-a)\Gamma(b)} \; & \; e^{-i\pi b} \frac{\Gamma(c)\Gamma(a-b)}{\Gamma(c-b)\Gamma(a)} \\ \; &\; \\ e^{-i\pi(a+1-c)} \frac{\Gamma(2-c)\Gamma(b-a)}{\Gamma(b+1-c)\Gamma(1-a)} \; & \; e^{-i\pi(b+1-c)} \frac{\Gamma(2-c)\Gamma(a-b)}{\Gamma(a+1-c)\Gamma(1-b)} \end{matrix} \right) \Phi_\infty$

where Γ is the gamma function.

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 $w = u v$ 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.

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

D k (λ,μ,ν; z) = s k (z)

is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.

Note that each triangle map is regular at z=0,1 and ∞ respectively, with

$s_0(z)=z^\lambda (1+\mathcal{O}(z))$

$s_1(z)=(1-z)^\mu (1+\mathcal{O}(1-z))$

and

$s_\infty(z)=z^\nu (1+\mathcal{O}(1/z))$

In the special case of λ, μ and ν real, with $0\le|\lambda|,|\mu|,|\nu|<1$ 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.

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 $\langle p,q,r\rangle=\Delta (p,q,r).$

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 $π 1$ 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.

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.

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