Heun's equation
Encyclopedia
In mathematics
, the local Heun function Hℓ(a,q;α,β,γ,δ;z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called a Heun function, denoted Hf, if it is also regular at z = 1, and is called a Heun polynomial, denoted Hp, if it is regular at all three finite singular points z = 0, 1, a.
ordinary differential equation
(ODE) of the form
The condition
is needed to ensure regularity of the point at ∞.
The complex number q is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β). Every second-order linear ODE on the extended complex plane
with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation
, can be transformed into this equation by a change of variable.
of the Coxeter diagram D4, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer.
The symmetries fixing the local Heun function form a group of order 24 isomorphic to the symmetric group
on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points. The complete list of 192 symmetries was given by using machine calculation. Several previous attempts by various authors to list these by hand contained many errors and omissions; for example, most of the 48 local solutions listed by Heun contain serious errors.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the local Heun function Hℓ(a,q;α,β,γ,δ;z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called a Heun function, denoted Hf, if it is also regular at z = 1, and is called a Heun polynomial, denoted Hp, if it is regular at all three finite singular points z = 0, 1, a.
Heun's equation
Heun's equation is a second-order linearLinear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
ordinary differential equation
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....
(ODE) of the form
The condition
is needed to ensure regularity of the point at ∞.The complex number q is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β). Every second-order linear ODE on the extended complex plane
Complex differential equation
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 points of the equation need to be studied...
with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation
Hypergeometric differential equation
In 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...
, can be transformed into this equation by a change of variable.
Symmetries
Heun's equation has a group of symmetries of order 192, isomorphic to the Coxeter groupCoxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
of the Coxeter diagram D4, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer.
The symmetries fixing the local Heun function form a group of order 24 isomorphic to the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points. The complete list of 192 symmetries was given by using machine calculation. Several previous attempts by various authors to list these by hand contained many errors and omissions; for example, most of the 48 local solutions listed by Heun contain serious errors.