Asymptotic behavior of Heun function and its integral formalism
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric , and functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrdinger equation of quantum mechanics, and addition of three quantum spins.
In this paper, applying three term recurrence formula Chou2012 , we consider asymptotic behaviors of Heun function and its integral formalism including all higher terms of ’s.111“ higher terms of ’s” means at least two terms of ’s. We show how the power series expansion of Heun functions can be converted to closed-form integrals for all cases of infinite series and polynomial. One interesting observation resulting from the calculations is the fact that a function recurs in each of sub-integral forms: the first sub-integral form contains zero term of , the second one contains one term of ’s, the third one contains two terms of ’s, etc.
Applying three term recurrence formula, we consider asymptotic behaviors of Heun functions and their radius of convergences. And we show why Poincaré-Perron theorem is not always applicable to the Heun equation.
In the appendix, I apply the power series expansion and my integral formalism of Heun function to “The 192 solutions of the Heun equation” Maie2007 . Due to space restriction final equations for all 192 Heun functions is not included in the paper, but feel free to contact me for the final solutions. Section 6 contains two additional examples using integral forms of Huen function.
This paper is 4th out of 10 in series “Special functions and three term recurrence formula (3TRF)”. See section 6 for all the papers in the series. The previous paper in series deals with the power series expansion in closed forms of Heun function. The next paper in the series describes the power series expansion of Mathieu function and its integral formalism analytically.
keywords:Heun equation; Three term recurrence relation; Asymptotic expansions; Integral formalism
Msc:33E30, 34A99, 34E05
rmkRemark \newproofpfProof \newproofpotProof of Theorem
The Heun function, having three term recurrence relations, are the most outstanding special functions in among every analytic functions. Due to its complexity Heun function was neglected for almost 100 yearsHeun1889 . According to Whittaker’s hypothesis, ‘The Heun function can not be described in form of contour integrals of elementary functions even if it is the simplest class of special functions.’
Recently Heun function started to appear in theoretical modern physics. For example the Heun functions come out in the hydrogen-molecule ionWils1928 , in the Schrdinger equation with doubly anharmonic potentialRonv1995 (its solution is the confluent forms of Heun function), in the Stark effectEpst1926 , in perturbations of the Kerr metricTeuk1973 ; Leav1985 ; Bati2006 ; Bati2007 ; Bati2010 , in crystalline materialsSlavy2000 , in Collogero-Moser-Sutherland systemsTake2003 , etc., just to mention a few.Birk2007 ; Suzu1999 ; Suzu1998 Traditionally, we have constructed all physical phenomenons by only using two term recursion relation in power series expansion until 19th century. However, modern physics (quantum gravity, SUSY, general relativity, etc) seem to require at least three or four recurrence relations in power series expansions. Furthermore these type of problems can not be reduced to two term recurrence relations by changing independent variables and coefficients.Hortacsu:2011rr
In previous paper we show the analytic solutions of Heun functions for all higher terms of ’s by applying three term recurrence formulaChou2012 ; power series expansions for an infinite and polynomial casesChou2012c .
According to Ronveaux (1995 Ronv1995 ), “Except in some trivial cases, no example has been given of a solution of Heun’s equation expressed in the form of a definite integral or contour integral involving only functions which are, in some sense, simpler. It may be reasonably conjectured that no such expressions exist.”
Instead Heun equation is obtained by Fredholm integral equations; such integral relationships express one analytic solution in terms of another analytic solution. More precisely, in earlier literature the integral representations of Heun’s equation were constructed by using two types of relations: (1) Linear relations using Fredholm integral equations. Lamb1934 ; Erde1942 (2) Non-linear relation (Malurkar-type integral relations) including Fredholm integral equations using two variables. Slee1969a ; Slee1969b ; Arsc1964 ; Schm1979
Now we consider direct integral representations of Heun functions and their asymptotic behaviors and boundary conditions for the independent variable by using 3TRF. Expressing Heun functions in integral forms resulting in a precise and simplified transformation of Heun functions to other well-known special functions such as hypergeometric functions, Mathieu functions, Lame functions, confluent forms of Heun functions and etc. Also, the orthogonal relations of Heun functions can be obtained from the integral forms.
In Ref.Heun1889 , Heun’s equation is a second-order linear ordinary differential equation of the form
With the condition . The parameters play different roles: is the singularity parameter, , , , , are exponent parameters, is the accessory parameter. Also, and are identical to each other. The total number of free parameters is six. It has four regular singular points which are 0, 1, a and with exponents , , and .Assume that has a series expansion of the form
We have two indicial roots which are and
2 Asymptotic behavior of the Heun equation
2.1 Poincaré-Perron theorem and its applications for solutions of power series
Let’s review certain theorems on the asymptotic behavior of solutions of linear difference equations with constant coefficients. Consider a linear recurrence relation of length with constant coefficients where
with . The characteristic polynomial equation of recurrence (2.1) is given by
Denote the roots of the characteristic equation (2.2) by .
H. Poincaré’s suggested that
Poincaré-Perron theorem Miln1933 : If the coefficient of in the difference equation of order be not zero, for , and other hypotheses be fulfilled, then the equation possesses fundamental solutions , such that
where and is a root of the characteristic equation, and by positive integral increments.
The recurrence relation of coefficients starts to appear by substituting a series into a linear ordinary differential equation (ODE). In general, the 3-term recurrence relation is given by
with seed values . For the asymptotic behavior of (2.3), where exists. Its asymptotic recurrence relation is given by
where . Due to Poincaré-Perron theorem, we form the characteristic polynomial such as
The roots of a polynomial (2.5) have two different moduli
In general, if , then , so that the radius of convergence for a 3-term recursion relation (2.3) is . And as if , then , and its radius of convergence is increased to . For the special case, is divergent when and , and it is convergent when . More details are explained in Appendix B of part A Ronv1995 , Wimp (1984) Wimp1984 , Kristensson (2010) Kris2010 or Erdélyi (1955) Erde1955 .
In chapter 3.3 on part A (pp. 34–36) Ronv1995 , they obtain three-term recursion system by putting a power series with an unknown coefficient into Heun’s equation about corresponding to the exponent zero. By applying Poincaré-Perron theorem, “We adopt the restriction and the series will generally have radius of convergence 1; it will therefore only represent a local solution.” And its theorem tells us that a Heun function of class I about , converging in the circle as is less than 1 where . Table 1 tells us all possible boundary conditions using Poincaré-Perron theorem.
|Range of the coefficient||Range of the independent variable|
2.2 Asymptotic behavior for an infinite series of and the boundary condition for
By rearranging coefficients of and terms in (1.3), let’s test for convergence of the Heun function about for an infinite series. As (for sufficiently large, like an index is close to infinity, or you can treat as ), (1.3)–(1.4b) are asymptotically equal to
Substitute (2.6b) into (2.6a) by letting .222We only have the sense of curiosity about an asymptotic series as for a given . Actually, . But for a huge value of an index , we treat the coefficient as for simple computations. For , it gives
If a series solution of a linear differential equation is absolutely convergent, we can rearrange of its terms for the series solution. Indeed, the sum of any arbitrary series is equivalent to the sum of the initial series.
With reminding the above mathematical phenomenon, let assume that a series solution of Heun’s equation is absolutely convergent. The sequence consists of combinations and in (2.7). First observe the term inside parentheses of sequence which does not include any ’s in (2.7): with even index (,,,).
When an asymptotic function , analytic at , is expanded in a power series, we write
Observe the terms inside parentheses of sequence which include one term of ’s in (2.7): with odd index (, , ,).
Put the above sequences in (2.10) putting .
Observe the terms inside parentheses of sequence which include two terms of ’s in (2.7): with even index (, , ,).
Similarly, the asymptotic function for three terms of ’s is given by
By definition, a real or complex series is said to converge absolutely if the series of moduli converge. And the series of absolute values (2.17) is
This double series is absolutely convergent for . (2.17) is simply
The coefficient decides the range of an independent variable as we see (2.19). More precisely,
|Range of the coefficient||Range of the independent variable|
where is the sufficiently huge positive real or complex. Then we can argue that for . For examples, if , then and as , .
In the case of assuming is huge numerical values, (2.18) turns to be
2.3 Original Poincaré-Perron theorem vs. revised Poincaré-Perron theorem
As we compare Table 2 with Table 1, both boundary conditions for radius of convergence are equivalent to each other since except . Table 2 allows for the analytic solution of a Heun function, but there is no solution for a series since in Table 1. As , their ranges of are slightly different: (i) Radius of convergence is the unity in Table 1 at , but we suggest that its radius is approximately 0.414214 in Table 2. (ii) If is quiet huge numerical real or complex values, their radius are almost equal to the unity, i.e., as in Table 2, its range approximates to , which is really closed to in Table 1. (iii) In the region at , maximum absolute value of in Tables 1 and 2 are quiet different. As we see where a positive real value in Table 2, it is a square root function of and the range of its slope with respect to is between 0.207107 and 1. A variable in Table 1 is just linearly increasing line with a slope 1 with respect to . Since is a negative real value in Table 2, the slope of a square root function of is between -1 and -0.207107. And the slope of in Table 1 is just -1. (iv) A square root function for a huge value in Table 2 is closed to which demonstrates strong justification of Poincaré-Perron theorem, but in the region at , Poincaré-Perron theorem is not available to obtain radius of convergence of Heun functions any more.
Now, let’s consider difference between Tables 1 and 2 with respect to numerical computations. A sequence is derived by putting a power series into a Heun’s equation. The boundary condition of in Table 1 is obtained by the ratio of sequence to at the limit . And radius of convergence of in Table 2 is constructed by rearranging coefficients and in each sequence .
First put (2.6b) in (2.6a) with and substitute the new (2.6a) into (2.21) by allowing with various positive integer values of in Mathematica program. Similarly, numerical values of with and are given in Tables 4 and 4.
Numerical values of in Tables 4 and 4 are derived by putting a 3-term recursive system into a power series with the specific values of and . As we see Table 4, is convergent as , its approximative value is . And Table 4 tells us that as is also convergent. It means that the radius of convergence using Poincaré-Perron theorem and the boundary condition by rearranging of its terms for the series solution are both available for the analytic solutions of Heun functions.
Consider the following summation series such as
A numerical quantities in Tables 6 and 6 are obtained by rearranging and terms in each sequence with the certain values of and . Table 6 tells us that is divergent as . And Table 6 informs us that as is also convergent which is equal to approximative quantities in Table 4. According to Table 6, we notice that the radius of convergence using Poincaré-Perron theorem is not available in an asymptotic series solution in closed forms which is performed by rearranging coefficients and terms in the sequence .
We can not use Poincaré-Perron theorem to obtain the radius of convergence for a power series solution. And a series solution for an infinite series, obtained by applying Poincaré-Perron theorem, is not absolute convergent but only conditionally convergent.
We might have curiosity why we have such errors since we apply one of any values in the interval of convergence of the series, constructed by Poincaré-Perron theorem, into asymptotic expansion by relating the series to the geometric series. To answer this question, first of all, consider an alternating harmonic series such as
This series is well known to have the sum since we add terms one by one. However, since we rearrange of its terms for the series solution, its sum can be divergent; if all terms are taken with + signs, it is divergent. Similarly, it is also divergent since we add all terms with -signs. This series is not absolutely convergent but conditionally convergent, based on the Leibniz criterion.
With reminding this example, let assume that a power series of Heun’s equation converges absolutely within its radius of convergence, obtained by applying Poincaré-Perron theorem. It tells us that even if we rearrange the order of the terms in series, its solution is also convergent. For instance, consider and as real positive numbers. is real positive number and is real negative one in (2.7). We observe that all terms in sequence in (2.7) consists of positive and negative real values; any terms having where are composed of real negative values, otherwise real positive ones. First, we take all terms with real positive values in each sequence in (2.7) and after that, take every terms having real negative ones. For with for simplicity, it gives
|Real positive terms||Real negative terms|
We construct a power series solution of real positive terms, denoted by and build a series solution of real negative terms, denominated by . Since we add and we get a asymptotic series .
(A) Series of
First observe the term of sequence which does not include any ’s of real positive terms in Table 7: with every index (,,,).
When an asymptotic series , analytic at , is expanded in a power series, we write
Observe the terms of sequence which include two term of ’s of real positive terms in Table 7: with every index except (, , ,).
Put the above sequences in (2.25) putting .
Observe the terms of sequence which include four terms of ’s of real positive terms in Table 7: with every index except (, , ,).
Similarly, the asymptotic series for six terms of ’s is given by
By mathematical induction, we repeat this process and build series solutions for all higher terms of ’s. We construct every terms where . Substitute (2.26), (2.28), (2.30), (2.31) and including all terms where into (2.24).
gives us a real positive value and the series of absolute values (2.32) is
This double series is absolutely convergent for and its boundary condition is same as Table 2. It informs that the radius of convergence of can not be obtained by applying Poincaré-Perron theorem
(B) Series of
Observe the term of sequence which include one term of ’s of real negative terms in Table 7: with every index except and (, , ,).
will be given by an expression
Observe the terms of sequence which include three term of ’s of real negative terms in Table 7: with every index except (, , ,).
Observe the terms of sequence which include five terms of ’s of real negative terms in Table 7: with every index except (, , ,).
And for seven terms of ’s is given by
provides us a real negative value and the series of absolute values (2.43) is