Heun functions, their generalizations and applications

The Heun Project


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The Heun functions

The Heun functions (named after Karl Heun:1859-1929) are unique local Frobenius solutions of a second-order linear ordinary differential equation of the Fuchsian type which in the general case have 4 regular singular points. Some of those singularities can coalesce and lead to Heun functions of the confluent type - confluent Heun functions, biconfluent Heun functions, triconfluent Heun functions and double confluent Heun functions.

The Heun functions generalize the hypergeometric function, the Lame function, Mathieu function and the spheroidal wave functions. Because of this, their application to science is significant: in the Schrodinger equation with anharmonic potential, in water molecule, in the Stark effect, in different quantum phenomena related with repulsion and attraction of levels, in gravitational physics of scalar, spinor, electromagnetic and gravitational waves, in crystalline materials, in 3d waves in atmosphere, in Bethe anzatz systems, in Collogero-Moser-Sutherland systems, e.t.c. Because of the wide range of their applications, they can be considered as the 21st century successors of the hypergeometric functions.

Despite the numerous applications, the theory of those functions remains far from complete. There are some analytical works on the Heun functions, but as a whole there are many gaps in our knowledge of those functions.

Particularly, the connection problem for the Heun functions is not solved - one cannot connect two local solutions at different singular points using known constant coefficients.

Another example of a serious gap in the general theory of the Heun functions in general is the absence of integral representations analogous to the one for hypergeometric functions.

Numerically, the only software package currently able to work with the Heun functions is MAPLE. Alternative ways for evaluations of those functions do not exist. The numerical realization in MAPLE however has some limitations which are hard to overcome considering MAPLE's closed kernel. Furthermore, outside of some specific cases, MAPLE's realization of those functions relies heavily on numerical integration, which however cannot be controlled by the user and thus it is much harder to interpret the results in the problematic areas.

All this led to the idea that a new numerical realization of the Heun functions is needed. This project is immense, since it requires work both on the theory of those function and finding a clever way to translate this theory into working numerical routines. Such a project however will find have a wide impact on science considering the numerous applications of those functions.

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