DIFFERENCE EQUATIONS
                                 & RECURSIVE RELATIONS  (1801-1825)








Difference Equations and Recursive Relations in this period have been studied by several mathematicians.  Some of these mathematicians include:
Babbage, Bessel, Farey, Gauss, Gompertz,and Legendre.
Babbage and the DifferenceEngine

  Ordinary arithmeticcalculating machines, in Babbage's day, could only carry out individualcalculations.They were useful only in the computation of single sums involvingaddition,subtraction, multiplication or division.  Charles Babbage worked on a calculating machine that could be adapted specially to performthe Method of Differences.  The Method of Differences is away of integrating"differences".    A"difference", written D f(x), is simply the difference between the values of a function taken attwodifferent values of its variable, x.  For example: Df(x)=f(x1)-f(x2 ). A "differential" is much the same thingexcept that the difference between the two valuesof x used, x1andx2, is reduced [gradually by a limiting process] by an infinitesimalamount, df(x). A "differential" is thus an infinitesimal "difference".Conversely, a "difference" can be seen as akind of finite"differential",one which uses an interval of a finite sizeinstead of an infinitesimal.

    The Method of Differences is a practical numerical technique. It can beused to calculate exactly the numerical values of any mathematical functionthat can be expressed as a simple polynomial .  Perceivingthe simplicity and facility of the Method of Differences andthe arithmeticoperationrequiredto execute it, Babbagerealised that its principles could be very easily embodied insimplemechanicalcomponents.  These insights led him, in 1821, to designanddevelopa DifferenceEngine.   This differenceengine only required mechanisms foradditionrepeated manytimes over to perform the tasks Babbage hoped for .  A clearer understanding of the mathematics of the Method of Differences is essential ifone is to understand Babbage'svision, and gain a better pictureof the mechanics of his creation, its operations,capabilities, limitations,andthe manner in which it was expected to achieveits object.

  Methods or algorithms for evaluating functions or solving equations by recursion,  ( the repetitive substitutionof the approximate solution of an equation backinto itself to obtain a betterapproximation),  had been devised asearly as the late 17th century.  Nevertheless, the Method of Differencesdidnot becomea clearly identifiable, branch of numericalmathematics untilsome time after Brook Taylorenunciated the famous theorem which bears his name ( Taylor's Theorem) during the 18th century.  By Babbage's time,however,the Methodof Differences had been in use for a long while and withmuch success forthe manual calculation of mathematical and other types ofnumerical tables.It was a well-tried and firmly established techniqueofNumerical Analysis,and its theory was well understood. What made it an idealprinciple for the automatic, mechanical computation of such tables was that,in its use, the same sequence of simple arithmetical operations is repeated overand over again. Therefore, a machine designed to imitate it only needed thosemechanisms for repeating these same operations.

  What makes theMethod of Differences a practical proposition for the interpolation of the numerical values of functions is a theorem which states that the nth differenceofapolynomial of degree n is a constant, and that all its differences of ahigherorder are zero and can therefore be ignored. As a result, when one isdealingwith the evaluation of polynomials, one need only work with a finite setofdifferences.  One starts with the assumption that one is tabulatinga polynomialfunction of the nth degree.  Consequently one may also assumethat itsnth difference is always a constant and having been given or havingascertainedstarting values for the function itself and its n levels of differencesonemay calculate from these, by adding them together in the correct orderrepeatedly,a full table of values for the function at the fixed and specifiedinterval,aprocess comparable to Integration.
  It will befound that this procedure can be used, in general, to generate in sequence,a full table of values for any function, for a given, fixed interval ofvalues for x. In principle all one needs is the starting values of the functionand its 1st to nth differences,and the assumption that the nth differenceis a constant. Once these are specified then a complete table of valuesfor any functionand all its differences can be produced by iteration. Ifone is tabulatinga polynomial function of the nth degree, then this willbe found to be anexact process, otherwise it will be found that it producesa seriesof reasonablysatisfactory approximations.

    The Difference Engine was only partially completedwhen Babbageconceived the idea of another, more sophisticated machine             called an Analytical Engine. Interestingly enough, more than one hundredand fifty years after its conception, one of Babbage'searlier        Difference Engines waseventually constructed fromoriginaldrawings by a team at London's ScienceMuseum. The device                 performedits first sequence of calculations in the early 1990's and returnedresultsto31 digits of accuracy, which is far more accurate     than thestandard pocket calculator.However, each calculation requires theuser toturn a crank hundreds,sometimes thousands of        times.  Babbageworked on his Analytical Engine from around 1830 until he died, but sadlyit was never completed. It is often said         that Babbagewas a hundred years ahead of his time and that the technologyof the daywas inadequate for the task.
 
 


Bessel Functions

Friedrich Wilhelm   Besselwas a German astronomer and mathematician,best known for making the firstaccurate measurement of the distance to astar.  He established theuniform system for computing star positions that is still in use.  In the investigation of problems connected with planetary perturbation,he introduced into mathematics the   Besselfunctions as the solutionsof certain differential equations. The functionsare of great importance in determining the distribution and flow of heator electricity through a circular cylinder and in the solution ofproblemsconcerning wave theory,elasticity, and hydrodynamics.
 
 

  Bessel'sdifferential equation is as follows:
  (d2u)/(dz2 ) + (1)/(z) * (du)/(dz)+ (1 - (s2/z2))u=0.         (1)

The Besselfunction can be defined as a particular solutionof Bessel'sdifferential equation (1).   Here, G is used to denote the gamma function (See Gaussbelow for information on the gamma function).  Using the properties of the gamma function,we get Bessel'sfunction to be:
 
 


Js(z) =  /(G(j+1)G(j+s+1))*(z/2)2j+s.          (2)
The sum of this series is a solution to (1) for any real s except if sisa negative integer.  If s is a negative integer, the series isnot defined.  This is because G(s) is infinite for negative integerss.  For a given value of s, not an integer, the functions J s(z)and J-s (z) give a pair of linearly independent solutions of(1).  Thus, allsolutions of (1) are AJs(z)+BJ-s(z), for arbitraryconstants A and B.  It is important to note, however,that  J-s(z) is infinite at z=0.
There are also identities that relate the Besselfunctions for differentvaluesof s.  They are as follows:
 

                    Js 1(z)=(s/z)Js (z) J'(z).                                 (3)
                    Js-1(z) + Js+1(z) =(2s/z)J s(z).                        (4)
The first pair of these recursion relations (3) follows from the seriesexpansion (2), while (4) follows from (3).
 

FareySeries

    The Fareyseries FN is the set of all fractionsbetween 0 and 1 written in lowest terms.  The denominators of thesefractions donot exceed N ( where N is the order of the series) and they arearranged inorder of magintude. In 1816, Farey published a statement in which he declared that the middle of any three successive terms in the Farey series is the median tof the other two.  It was Cauchy who eventually supplied the proof of this statement.
A proof of this statement is as follows:
    Start with F1 which is 0/1,1/1.  In order toget F2, insert the mediant intoF1 :0/1,1/2, 1/1.For F3 we add two mediants: 0/1,1/3,1/2,2/3,1/1. To get F4 we also add only 2 fractions: 0/1,1/4,1/3,1/2,2/3,3/4,1/1.  Next, add 4 fractions to get F5:0/1,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,1/1. The general rule is this: to move from FN-1 to F N add all possible mediants (that come out to be in the lowestterms)with N in the denominator. Since forming a mediant may only increasethe denominator weare led to think that following this rule we indeed willget the wholeof FN. To complete the proof we refer to the Stern-Brocot   tree which contains all positive fractions.Soin the process of constructing a Fareyseries no fraction will be missed.

When N is prime, the rule adds N-1 fractions. In general,  (N) fractions are added. For all reducible fractions m/N will have appeared in one of the earlier series.

The Fareyseries is important in theproof of a corollary of Euclid'salgorithm.  The algorithm is as follows:
For integers m and n with gcd(m,n) = 1 and m n, there exist positive integers a and b such that ma - nb = 1. The proof  relies on the properties of the Stern-Brocot    tree. For any two consecutive fractions m1/n 1  and m2/n2 in the Fareyseries, m2n1- m1n2=1. So,depending on whether m or n is larger, locate either m/n or n/m ina Fareyseries and select (asa/b) eitherthe precedingor the following fraction.

Gauss and the Gamma Function

The gamma function is useful more for its relationship to other functions than as a solution by itself of some problem. It arises in simplifyingthe evaluation of some infinite or improper integrals and in the solutionofdifferential and difference equations arising in probability theory,statistics,mathematical physics, and engineering mathematics.

 The logarithm can be defined by and integral.  The gammafunction can also be defined by an integral, but in a different way. It is:
 


  for 0<x<infinity .

 The integral converges because the exponential makes it smallass-> infinity .  As s->0, thefactor s^x-1 is integrable becausex>0.
The gamma function is a generalization of the factorial.  In fact, it has the following properties:
 


G (x+1)= x G(x).
  G(n+1)= n!  ,  if n is a positive integer.
    G(1/2) = 

Another useful identity with respect to the gamma function is:
 



G(x) = (2x-1)/( ) G(x/2)G(x+1/2).

 This can be used to give another derivation of the gamma function on the "half-integers."

The formula for the surface area of the sphere of radius R in n dimensions is:
 
 


An=( 2 pn/2 Rn-1 )/G(n/2).
Gompertz Equation
Benjamin Gompertz  applied the calculus to actuarialquestions.  He is best remembered for Gompertz's Law of Mortality in which he showedthat the mortality rate increases in a geometric progression.Hence, whendeathrates are plotted on a logarithmic scale, a straight lineknown as theGompertzfunction is obtained. It is the most informative actuarialfunctionfor investigating the ageing process. The slope of the Gompertzfunction lineindicates therate of actuarial ageing.
The Gompertzgrowth law has been shown to provide a good fit for the growth data ofnumerous tumors.  It can be described by the following system of differentialequations:

Let V(t) measure the size of the tumor (e.g. volume, number, etc.)
 
 


 Here  tumor growth,
retardationconstant,
initialgrowth or regression rate.

The sign of beta determines where the tumor grows or regresses. The solution is given by:


where the constants are determined by the data.
In a serial treatment model, it is assumed that the post-treatment tumorhas the same growth characteristics as the original tumor.  Specifically, it is assumed that the ultimate tumor size remains unchanged.  We denote this as:
 

The Gompertz growth model can be written in the following form:
 

To determine the tumor regression from just before one treatment to justbefore the next, we integrate the first order differential equation as follows:
 

This leads to thedifference equation:

Legendre Functions

In problems of spherical symmetry, one encounters Legendre'sdifferentialequation:
 
 


[(1- z2)u']'+yu=0.

Here y= l(l+1) for some integer l >=0.

This equation is easily solved by the power series:
 


u(z) =  zk.

 When we substitute the power series into Legendre's differentialequation, we get:
 
 


k(k-1) zk-2 -  (k2+k-y)zk=0.

 We next replace k-2 by k in the first sum.  The coefficients of the like powers of z must match, so that:
 
 
 


ak+2=ak (k(k+1)- y )/((k+2)(k+1))   ,(k=0,1,2....)
 
It follows that the Legendrepolynomials are:
 

Pl(z) = 1/2l. /(j!)*((2l-2j)!)/((l-2j)!(l -j)!)*zl -2j.

(m=l/2 if l is even, and m=(l-1)/2 if l is odd.
The Pl(z) satisfy the recursive relation:
 

(l+1)Pl+1 (z)-(2l +1)zPl(z)+lPl-1 (z)=0.