Frobenius Series Solution of Fuchs Second-order Ordinary Differential Equations via Complex Integration

A method is presented (with standard examples) based on an elementary complex integral expression, for developing Frobenius series solutions for second-order linear homogeneous ordinary Fuchs differential equations. The method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method is straightforward to apply as an algorithm, and eliminates the manipulation of power series, so characteristic of the usual approach [14]. The method is a generalization of a procedure developed by Herrera [4] for finding Maclaurin series solutions for nonlinear differential equations.


Introduction
We consider the Fuchs second-order linear ordinary differential equation (ODE)

W. Robin
with the superscript numbers in brackets denoting differentiation with respect to , z the zeroth derivative being the function ) (z f itself.As usual, we assume [14] ) are analytic functions of the independent variable , z with 0 z being a regular singular point of (1.1)..The class of linear ODE represented by equation (1.1) contains many of the important equations of classical Mathematical Physics and its solution is of sufficient practical importance to warrant further study.
The usual [6,14] Frobenius power series solution for the linear ODE (1.1) is with the unknown coefficients } { m m a and the index r to be determined by substituting (1.3) into (1.1), which gives rise to a recurrence relation for the In particular, the index r is obtained as the solution(s) of the indicial equation (1.4) which arises form the leading term ( 0 0 ≠ a ) on substituting (1.3) into (1.1)[6,14].This, the standard approach, involves (in general) the manipulation of infinite series.However, by an elementary device, the necessity for manipulating infinite series directly, when solving linear ODE, can be side-stepped and, in the resulting format, manipulations are reduced to solving simple equations along with some basic algebra.The basis of this alternative approach to the determination of Frobenius power series, is the well-known formula from complex variable theory [5] (all closedcontour integrals that occur below are assumed evaluated in the counter-clockwise or positive direction) where z is a complex variable and 0 z a fixed point within the closed-contour .C The relation (1.5) is used to 'knock-out' all but one term from (1.3) through a combined differentiation/integration process, described in detail in section 2, which leads to a complex integral representation, equation (2.2) below, for the coefficients of (1.3).

955
Once equation (2.2) is established, it becomes a routine matter to apply it to the solution of wide classes of ODE, especially those of the Fuchs' class that are the main topic of discussion here.Assuming a solution of the Frobenius form n ) From the operational point of view, the basic problem of solving the ODE is reduced, in practice, to the repeated solution of a simple algebraic equation in one variable!From the historical point of view, the method is a generalization of a method originated by Herrera [4] for dealing with Maclaurin series solutions to nonlinear ODE.
The path of the paper proceeds as follows.In section 2 we derive the basic formula, (2.2), as a generalization, to the case of a Frobenius power series, of Herrera's formula for the coefficients of a Maclaurin series [4].Next, in section 3, we apply the algorithm to some standard ODE from mathematical physics: the Bessel equation of order ν [1,14], the hypergeometric equation [5,14] and Heun's equation [2].In these examples, and in the others that follow in sections 4 and 5, we consider the problem solved as soon as we have developed the recurrence relation, the rest of the solution process being well-documented in the literature.In section 4 the Riemann-Papperitz equation is considered and we see, explicitly, the effects of the standard transformation approach [3] to its solution on the transformation of the recurrence relations for the coefficients.Finally, section 5 provides a brief discussion of the method (its generic problems and other applications) and 'round things off' with a another couple of examples.
Note that, while we have applied the method to a standard range of 'classical' equations; many other equations will, naturally, yield a solution to this approach.Further, we have restricted ourselves to the consideration of an individual singular point in each example below: the singular point at the origin.Other singular points of the ODE may be transformed to the origin, if necessary.

Derivation of the Basic Formula
Progressing formally, suppose we start with an assumed Frobenius power series expansion of a function, ), (z f a regular singular point ,

W. Robin
for constant index .
r Then, with C a closed-contour around 0 z avoiding other singularities of ), (z f we wish to show that, for , , where, following the notation of Ince [6], for positive integers k First, we differentiate (2.1) k times, to find that Next, if we divide through (2.5) by 1 0 ) ( and integrate round a closedcontour , C containing 0 z while avoiding singularities of ), (z f we get, as the index r cancels-out or, changing dummy variables, we have (2.2).Finally, looking back, we find , , becomes the original formula of Herrera [4], that is We move on, now, to some standard examples.We note that series convergence is discussed, e.g., in [6], [7] and [14] and is not considered here.

The Solution of Some Standard Second-Order Equations
In this section, we will restrict the discussion to equations with regular singular points at the origin.Notice that we will always write the general recurrence relation in terms of , n a which is as it must be written [6,14].As a first example, we solve the Bessel equation of order , Next, Assuming (2.1), we divide through equation (3.1) by and integrate round the closed-contour , and compare the powers of the denominators of the integrands of (3.2) with that of (2.2) to get four equations for the dummy-variable , m one for each value of k (two, one, zero and zero, respectively), that is Having identified the appropriate values of k and , m we use (2.2), again, to rewrite equation (3.2), after cancelling and re-arranging, as which is the indicial equation, with solutions .The third and final step is to solve the recurrence relation (3.4) and obtain the Frobenius series solution explicitly.As this is well-known [1,6,14], we stop here Now, before we consider our next example, we pause to point-out that the index r has, once more, cancelled-

W. Robin
out of the dummy-variable ( m ) calculations.This appears to be a characteristic of the method as a whole (pace section 2).
For our second example, we solve the hypergeometric equation, that is [5,14] where , α β and γ are constants.As before, assuming ( and compare the powers of the denominators of the integrands of (3.7) with that of (2.2) to get five equations for the dummy variable , m one for each value of k (two, two, one, one and zero, respectively), that is and, in this case, the indicial equation is 0 Apparently, the first root, , 0 = r will yield an ordinary power series.With 0 0 ≠ a an arbitrary constant, we recognize, in (3.9), the recurrence relation(s) for the Frobenius series solution of the hypergeometric equation [5,14] and, as before, we terminate the example here.
As our final example in this section, we consider the Frobenius series solution of Heun's equation, that is [ and q constants and where . 1 Clearing fractions' in (3.11), we then multiply-out and collect like terms to get Next, assuming (2.1), we divide through equation (3.12) by

∫ ∫ ∫
and compare the powers of the denominators of the integrands of (3.13) with that of (2.2) to get eight equations for the dummy variable , m one for each value of k (two, two, two, one, one, one, zero and zero, respectively), that is Having identified the appropriate values of k and , m we use (2.2), again, to rewrite equation (3.12), after cancelling and re-arranging, as and where we have used the fact that . 1 In this case, from (3.15b), when 0 = n we must have ; 0 Again, the first root, , 0 = r will yield an ordinary power series.Equation (3.15) constitutes the recurrence relation(s) for the Frobenius series solution of the Heun equation (see [2] and references therein) and, as usual, we terminate the explicit calculations here.However, we see that we have a three- term recurrence relation, (3.16), for the Heun Frobenius series solution.For a brief discussion of this matter of three-term recurrence relations see Figueiredo [2].Further, if we set 1 = n in (3.15b), we find 1 a given in terms of 0 0 ≠ a and we have, indeed, only one solution.

The Riemann-Papperitz Equation
In this section we consider the Riemann-Papperitz equation and its relation to the hypergeometric equation [3].To facilitate the discussion, we consider, first, the following.The indicial equation has its own defining differential equation 0 To see this, we apply the method, with , 0 = n to (4.1), when we find that (4.1) transforms to

Frobenius series solution
which is, indeed, the indicial equation for (1.1).The equation (4.1) is just Euler's second-order homogeneous equation and is obtained (asymptotically) from (1.1) when (1.2) is rewritten as and then the leading terms are 'extracted' as .0 z z → We are in a position, now, to examine the Riemann-Papperitz equation, that is

W. Robin
If we apply the method to (4.7), for the regular singularity at the origin ), 0 ( = a then we find, after cancelling and re-arranging, that (4.7) transforms to As a further consistency check on (4.8), without actually writing out the series, we note, following [3] again, that (4.7) reduces to the hypergeometric equation, (3.6), 'after reduction', when we choose , and In this reduction process, we expect (4.8) to 'reduce' to (3.9) and this is the case.Further, making the substitutions (4.10) in (4.8), we find that, after some algebra also, and we have retrieved the recurrence relation (3.9b) for (3.6), as required.

Conclusions and Discussion
While the modern trend is for solving ODE using computer algebra systems [13] (which is perfectly natural and necessary in most cases) the facility with which the

W. Robin
As for the determination of the 'second solution' to (1.1), we have, at least in principle, the general solutions to our equations through the 'reduction of order' method [14].Alternatively, the special 'derivative method' for finding a second solution (given a Frobenius series solution), as explained, for example, in the standard textbook of Rainville [12].can be attempted.Or, again, it is possible to derive an ODE for the 'series part' of the known form [14] of the second solution and solve this using the present method.The actual choice of which method to attempt will depend on the particular circumstances and the details of the 'first solution'.
Naturally, the method presented here can be applied to higher-order ODE.Suppose we consider the example of the ℓ th -order linear homogeneous ODE with four singularities analysed, recently, by Kruglov [10].that is and we have Kruglov's scheme [10], so we finish here.Finally, we wish to draw attention to a certain 'family relationship' between (2.2)/(2.7)with , 0 = r and the Caputo fractional derivative [8]  (5.6) though (5.6) is applied to different types of equations and in a different way [8].

( 1 . 3 )
, one multiplies through the given differential equation by , integration (round a particular closed-path) throughout the resulting expression and applies equation (2.2), term by term.The result of this process, is that the original differential equation is transformed into the recurrence relation for the coefficients, tout court.(The process, or algorithm, transforms the dummy variable m into the dummy variable .
, we recognize (3.4) as the recurrence relation(s) for).(z J ν appropriate values of k and , m we use (2.2), again, to rewrite equation (3.7), after cancelling and re-arranging, as 0 points at b a , and , c but we will seek a Frobenius series solution to (4.5) by first moving the regular singular points to 1 , 0 and .∞ The argument is a standard one[3] and involves the determination of the s A' and s B' in terms of the roots of the indicial equations corresponding to the three regular singular points at b a , and .cSo, if the roots of (4.3) corresponding to a z = are denoted by , Papperitz equation may be rewritten, on clearing fractions, as[3] relation (4.9) provides a consistency check on the calculation, as, with ,