A New Perspective on the Foundations of Analysis

7/31/2019 A New Perspective on the Foundations of Analysis

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A N ew Perspect ive on the Foundations of A nalysis

A New P erspect ive on the Foun dat ions of Analysis

by Chappell Brown

Author : Chap pell Brow n First created : June , 1993 Last revision :February, 1996 Page: 1

Revised version of a pap er origninally p ublished in Physics Essays , Sept. 1995


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ABSTRACT

T.E.Phipps, Jr.s new definition of discrete infinite

pr ocess valu e is exam ined in th e light of asymptotic

analysis. The analysis suggests a more genera l

approach to limit operations, which are

conventionally used to define these values. Ananalogy between imaginary num bers and divergent

processes is proposed, which helps to explain the

current state of analysis, where man y infinite

processes lack numerical content due to problems

created by divergence. A general method whereby

any real function can be resolved into convergent

and divergent parts, just as complex numbers are

resolved into real and imaginary parts is described.

By defining a generalized limit as the limit of the

convergent part of a function, a path to a fully

general formulation of limits that would finallybanish d ivergence from analysis is described.

Key Word s: divergent series, sum mability, term inal

summation, asymptotics, numerical methods, set

theory.




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Resume

La nouvelle definition don nee p ar T.E. Phipp s au

traitemen t d es valeurs d iscretes et infinies est

examinee par la voie de lanalyse asymp totique.

Cette analyse prop ose une method e general pou r

traiter les limites qu i sont utilisees pou r la

definition des valeurs. Une similitud e entre les

nombres imaginaires et la d ivergence du processus

est proposee. Cette dern iere va aider a expliquer

letat actuel de lanalyse on p lusieur tr aitements

manqu ent le contenu n um erique p ose par le

probleme de divergence. Une method pour unecomp lete et generale formulation d es limites qu i

pou rraient finalement faire d isparaitre la

divergence est decrite.

1. In tr odu ction

After more th an th ree cent ur ies of developmen t, it might s eem th at little could be

added to established methods for assigning values to infinite sums, and yet

T.E.Phipps, J r. a sks u s to reconsider t he funda ment al definition, first form alized in

th e ear ly 19th cent ur y by Cau chy an d Bolzan o wher eby the value of th e expression

a0+ a1+ a2+ (1)




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is obta ined as t he limit

limn

{a 0+ a1+ a2+ + a n -1} (2)

when it exists.

This definition h as a n int uitive feel tha t gives it a cert ain validity even outside th e

str ict r ealms of ma th emat ical rigor. It is grounded in t he pr actical observat ion t ha t

when ever a s equen ce of valu es ap pr oaches closer a nd closer t o a sin gle fixed valu e,

th e process itself conta ins t he fixed limit a s its un ique, ultima te a nd tr ue value.

This implicit definition seems to have been assumed without any formal definition

from t he st ar t. Cau chys contr ibut ion wa s to explicitly form ula te a genera l definition

of the limit of a function, a nd to ma ke it a corn erst one of a fully axioma tic

developmen t of th e calculus. The definit ion of th e limit of an infinit e sum wa s th en

derived a s one special form of this limit .

By now, th is definition of th e value of an infinite su m a s t he limit of par tia l finite

sum s is t her efor ba cked both by ba sic int uition a nd th e force of rigorous

ma th ema tical deduction, giving it at th is point in th e developmen t of science, an

au ra of na tu ra lness. Indeed, how else would one define such infinite pr ocess values?

2. Term inal Sum ma tionPh ipps begins with th e tru ism tha t, for a ny value ofn , we have

V = {a 0+ a1+ a2+ + a n -1}+ { a n + a n +1+ a n +2+ }

= S n + R n (3)

A revolut ion in our view of infinite su ms occurs wh en we begin t o as k ques tions

about t he remainder term R n . If in fact th e sum is convergent a ccording to th e

Cau chy definition, then it is clear t ha t Rn ten ds to zero as n ten ds to infinit y. We

only need to shift the pa rt ial sum t erm t o th e left-han d side of th e equat ion t o see this,




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since th e part ial sums a re getting closer to V as n becomes lar ge. But suppose th e

sum is not convergent in t he st an dar d sense. Phipps identity now begins t o take us

towar d new r esult s, since we ha ve the option of explorin g the a sympt otic beha vior, a s

n ten ds t o infinity, of th e well defined finite p rocess

V -{a 0+ a1+ a2+ + a n -1} (4)

An example ma y serve to illustr at e how an a symptotic estima tion of th e rema inder

can yield a finite valu e for a divergent sum . If we set an =1/(n +1) , then we have

th e ha rm onic series, whose part ial sum s slowly diverge to infinity. However, th e

ratio

V - 11

+ 12

+ 13

+ + 1n

- log (n ) (5)

tends t o un ity, no matt er what value is assigned to V. This suggests th at we can t ake

-log(n) as a n a symptotic estimat e ofRn as n tend s to infinity. We now can sta te

that

V = a0+ a1+ a2+ + a n -1 +

= l imn

11

+ 12

+ 13

+ + 1n -1

- log (n ) =0.57721. . . (6)

In th is case, n orm ally considered divergent an d t her efor devoid of mean ingful

inform at ion, we find th at an asympt otic estimat ion of th e rema inder brings th e

process back into the rea lm of convergence. Becau se th is appr oach t o infinite

summ at ion uses an estimat e of th e remainder, or t erminu s of a sum , Phipps named

it ter mina l sum ma tion. In fact, th e first definit ion of th is sort was a pplied to

cont inued fra ctions (1) , which P hipps th en extended to summ at ion of series and a

var iety of oth er discret e infinite processes. In a r ecent s eries of paper s, the

applicat ion of th e met hod ha s been a pplied to both convergent an d divergent




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processes (2), (3), (4). One clear victory for th is summ at ion app roach, described in (3)

was t he direct comput at ion of th e sum

1+ 1x + 2x 2+ 6x 3+ + n !x n + (7)

conventionally divergent for all non-zero values of the variable x , at enough points to

graph th e fun ction it r epresent s. A var iation of this expression wa s first considered

by the 18-th centu ry Swiss math emat ician Leonh ar d Eu ler , and oth er versions h ave

cropped u p in discussions of divergent series. Since th ere is no kn own m eth od, or

even a r at iona le for dir ectly comput ing such a ser ies, an alysts in var iable fall back on

form al ar gument s. One tr eats t he series as th ough it were absolut ely convergent,

an d th en derives some properties which lead t o alter na tive means for deter mining its

valu e. For examp le, if (7) is su bst itu ted for f in th e different ial equat ion

1+ x ddx

xf (x ) = f (x ) , f (0)=1 (8)

an d t hen form ally manipulat ed as if it were convergent, we will find t ha t it

identically sat isfies the equa tion. Such form al met hods have been ta ken t o a h igh

degree of perfection, an d one n ow spea ks in ter ms of form al calculi, th ere bein g a

nu mber of different rout es to crea tin g form al versions of th e calculus (5) . But despite

th e distinction between form al an d convergent ser ies, Ph ipps shows th at his

ter mina l summ at ion pr ocess actua lly reproduces th e values of th e real valuedfun ction wh ich is a solut ion t o this equat ion - an import an t indicat ion t ha t t his

approach t o summ ation is on the r ight tra ck.

At t he sam e time, a large and a ctive field of ma th emat ics ha s grown up a round

attempts to circumvent the restriction imposed by the standard definition of

convergence by defining a d h oc summ at ion met hods (6),(7). In t his field, called

sum ma tion th eory one begins with a process t ha t converges for convergent sum s,

but a lso converges for certa in classes of divergent su ms. The extended m eth od of

sum ma tion agrees with th e convergent limit of par tial sum s, when it exists, and

th erefor in some sense represent s a n extens ion of th e notion of infinite su mma tion.

When problems of divergence crop up in other bra nches of ma th emat ics, we find




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corr espondin g notions th at r educe the process to a finit e value. Thu s, in t he th eory of

distribut ions, divergent int egrals ar e sum med by extra cting th e Ha dam ar d Finite

Par t(8) . We could cont inu e cat aloguin g the diverse met hods th at ha ve been a pplied,

often with much ingenuity, to repairing the divergent fissures in the edifice of

modern m at hem at ics. But t o th e novice, or t o physicist s who only wish to apply

mathematics to elucidate the connections between observable quantities, such a

cat alogue would seem str an ge indeed. Since mat hema tics purports t o be an

economical system with only the m inimum of hypoth esis and th e rest derived

th rough t he a pplicat ion of logic, we might wonder wha t lies at th e root of th is

mu ltiplicat ion of individua l meth ods, each one fabricated for a special problem.

It is here t ha t we mu st give credit t o Ph ipps for enu nciat ing a fun dam enta l

principle. He argues(9)

th at divergence itself is an a rt ifact of th e restr icted na tu re ofCauchys definition. Rath er th an r epresentin g some inevita ble barr ier tha t a ny

form ulat ion of ma th ema tics will encoun ter , he maint ains t ha t divergence will

simply disappea r wh en t he ba sic form alism is modified with a st ronger definition.

The rem ainder estima tion appr oach, which a llows discret e infinite pr ocesses to

converge, even when th ey diverge accordin g to the a ccepted definition, represen ts a

new, more r obust , definition of convergence. Extending t his a rgum ent , he claims ,

will event ua lly erad icat e divergence from ma th emat ics.

While th e meth od ha s ha d success in th e initial steps of such a pr ogram , this

repr esent s a bold claim. The read er might well wonder wh y, if such a program h ad

an y cha nce of success, it would not h ave alrea dy been t ried. We can look back on

severa l centu ries of brilliant work, where t he best ma th emat ical minds h ave

sear ched diligent ly for t he foun dat ions of an alysis, an d add t o th at th e profoun d

investigat ions of 20th cent ur y mat hem at ics int o th e logical foun dat ions of

ma th ema tics. With th is body of work a t ha nd, it would seem th at a divergence-free

version of the calculus could easily be realized, if it were at all possible.

But per ha ps th at is the crux of th is issue. If one believes a t ask is impossible, th en

an y set of tools an d techniqu es, no mat ter how powerful, will be of no avail. Thebarr ier is not th e problem itself, but r at her th e belief th at it cannot be solved. So the

idea th at th ere is a na tu ra l boun dary between convergent a nd divergent processes

is th e point t ha t n eeds to be exam ined, not so much a s a logical pr oposition, but




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ra th er as a psychological issue. It is th at belief th at Ph ipps has questioned with h is

app roach t o finding t he va lue of discret e infinite pr ocesses.

3. An Ana logy with Im aginar y Numb ers

The cur rent situa tion h as a par allel in th e history of ma th emat ics. When we reflect

on a ter m su ch a s an imagina ry nu mber , we begin t o perceive a similar a mbiguous

situation, and perhaps a recurr ent ha bit of mind in regard to mat hemat ical enti t ies.

For centu ries, math emat icians r egar ded symbols such a s - 1 as inherently

contradictory, although they recognized that meaningful results could be achieved by

ma nipulating t hem. Such a symbol was viewed as h aving no int r insic numerical

value, but could n evert heless be man ipulated in a pur ely form al ma nn er t o obtain

valid resu lts t ha t could be justified independ ent ly, usu ally by a more complex rout ethat had the virtue of using only acceptable real numbers.

Today, the sam e langua ge and meth ods a re a pplied to an object su ch a s (7), which,

contr ar y to a similar stru ctur e such a s

1+ x + x2

2+ x

3

6+ + x

n

n !+ (9)

considered t o have a d irect n um erical m eanin g, is viewed as ha ving only a sha dowy

existence as aform al st ru ctur e. We ar e allowed to ma nipu lat e (7) as th ough itwere on a n equal sta tu s with (9) but m ust bear in mind th at su ch divergent series

really have no true n umerical mean ing, and results obtained via such a short cut

mu st be later justified by valid convergent processes.

By directly summ ing both of th ese express ions st ar tin g with th e sam e definition of

infinite process value, Ph ipps denies us t he critical distinction bet ween conver gent

an d divergent pr ocesses on which such think ing is based. It tu rn s out t ha t th ere is

no logical basis for viewing some infinite processes as having a more numerically

valid sta tu s th an oth ers. And t he a na logy here between convergent/divergent series

on t he one ha nd, and r eal/imaginar y numbers on t he other tur ns out to have a

deeper mea ning. Indeed, we will now show tha t gener al classes of fun ctions can be

set u p which, in an alogy to complex nu mber s, can be resolved int o conver gent (rea l)




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an d divergent (imaginar y) par ts. A convergent pr ocess will then simply be a

par ticular case wh ere t he divergent par t is zero.

For t he sa ke of ar gumen t, we will consider t he m ore genera l cas e of fun ctions of acont inuous va riable,x as x ten ds t o infinit y, from wh ich we can d erive th e special

case of discrete infinite processes by allowing x to take positive integer values.

Before going int o the pa rt icular s of such a resolut ion, we can gain some insight into

th e larger pictu re by assu ming th at we have a form al definition of convergent a nd

divergent pa rt s, ana logs of the real a nd ima gina ry par ts of a complex var iable. For

example, we are accust omed t o writin g z = x + iy where x = Re (z ), the real part

of z an d y = Im (z ), or t he imaginary part ofz . In a similar vein we set

f = Cvt(f )+ Dvt(f ) , wher e Cvt(f ) represents th e conv ergent partoffand Dvt(f )

the divergent partof th e fun ction. Rat her t ha n nu merical quan tities, Cv t(f ) an d

Dvt(f ) are specific fun ctions a ssociat ed with f and ar e best th ought of as operators

ma pping an y fun ction f ont o cert ain subclasses of th e set of real valued fun ctions.

Since we would like to say t ha t th e divergent par t of a convergent fun ction is zero, it is

evident tha t Cv tma ps th e rea l fun ctions ont o th e set of conver gent fun ctions, which

we will denote by C. The new element in t his form alism is th e operat or Dvt and

specifying which set of divergent fun ctions sh ould const itu te its r an ge lies at t he

hea rt of the issue before u s. Denoting t his set of fun ctions by D, we can conceptu ally

cha ra cter ize it a s th ose fun ctions wh ich a re pur ely divergent just a s th e imaginar y

axis of nu mbers yi , is th e set of pur e imaginar ies.

An import an t differen ce here is th e lack of a simple divergent ba sis quan tit y

ana logous t o the pur e imaginar y i . Eu ler at tempt ed to deal with divergent

processes in just t his way, by int roducing an infinite quan tity int o analysis, which h e

symbolized by th e lett er i, and m an aged to derive some valid results u sing algebra ic

ru les defined for it. However, th is appr oach cont ain s logical flaws , which wer e only

ironed out in t his cent ur y with Abra ha m Robinsons invention of Non-Sta nda rd

Ana lysis. Although we do not ha ve a simple basis qua nt ity an d consequent algebra ic

system for representing functions, the assumption of a resolution into two classes of

fun ctions, one convergent a nd t he other divergent.can still be ma inta ined as a basisfor th e forma lism.

We mu st ma ke one special pr ovision for th e zero fun ction, which we will assum e




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belongs t o both convergent an d divergent classes. This exception is required for

cons isten cy with t he existin g definit ion of conver gence. We want to be able to say th at

when f is a convergent fun ction, its divergent par t Dvt(f ) is ident ically zero for all

x even th ough th e zero function is t echn ically conver gent.

Algebra ic ma nipula tion of the form al r epresent at ion of a function a s a sum

convergent a nd divergent pa rt s yields some genera l ident ities. The r eader s hould

ha ve no problem pr oving th e following, which ar e direct cons equen ces of th e un ique

decomposition of a fun ction int o convergent a nd d ivergent par ts:

Cvt(Cv t(f )) = Cv t(f )

Dvt(Dvt(f )) = Dvt(f ) (10)

Cv t(Dvt(f )) = Dvt(Cvt (f )) =0

These ident ities ma y seem tr ivially obvious, but oth er r elationships we might expect,

based on experience with complex num bers, can not be der ived without additiona l

assu mpt ions. For example, when two complex numbers ar e summ ed, the r esult is

foun d by adding th eir real an d imaginar y part s. Now if we assum e tha t t he sum of

an y two members ofD also belong to D, th en t his a ddition th eorem also holds for our

represen ta tion of real functions. This fact is immediately evident wh en we compa re

th e two represen ta tions of a su m of two fun ctions:

f + g = Cv t(f )+ Cv t(g )+ Dvt(f )+ Dvt(g )

(11)

f + g = Cvt(f + g )+ Dvt(f + g )

If th e sum Dvt(f )+ Dvt(g )belongs t o D, then since the divergent pa rt off+ g ,

Dvt(f + g ), is un ique we have the equa tion Dvt(f + g )= Dvt(f ) + Dvt(g ).

Reasoning along these lines, we can show tha t t he operator Dv tis linea r if and only

if th e set of functionsD is closed un der t he operat ion of add ition. Can we safely mak e

th is assum ption? Operat ing under th e most conser vative program , we will refra in

Author : Chap pell Brow n First created : Jun e , 1993 Last revision :February, 1996 Page: 10



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from mak ing such genera l assum ptions unt il we have tried out th e theory on

par ticular cases. There is nothin g in th e assu mpt ion of th e un ique decomposition we

ar e proposing here th at implies th e linear ity of th e set D. But it will tur n out a s we

explore t he det ails of specific classes of functions, t ha t t his a ssu mpt ion is t ru e for

rest ricted su bsets of real fun ctions. The set of fun ctions which rema in bounded a s x

ten ds t o infinity is one example of a subset of fun ctions for which th e operat or Dvt is

l inear .

We can summ ar ize what we have lear ned about t he linearity of addition with a

theorem:

Theorem 1

Let L denote a linear subspa ce of th e rea l fun ctions such th at th e set

Dvt(f )| f L is also a linear space, then th e operat ors Cv tand Dvtare l inear

when restr icted toL .

We have included th e opera tor Cv t in this theorem but at the sam e time it is

importa nt t o realize tha t a su btle asymmetr y subsists between these two operators.

Fir st of all, the set C of convergent fun ctions, a na logous to th e rea l nu mbers, is a

linear spa ce. But t his fact does not guar an tee th e linear ity of th e opera tor Cvt for

th e following reason. It is fairly element ar y to show th at , assum ing only the u nique

decomposition of fun ctions int o convergent an d divergent pa rt s, th at Cvt is linear if

an d only if Dvt is linear. Since we have proved th at th e linear ity ofDv tdepends on D

being a linear spa ce, th en it follows th at th e linea rity ofCv talso depends on th e

linear ity ofD , not C.

This odd resu lt is one example of th e potent ial pitfalls of proceeding on a pur ely

form al basis. But t hese int roductory rem ar ks do help to provide some orient at ion

that will be valuable as we delve into the technical details of actually effecting a

resolut ion of rea l valued functions.




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4. Resolution of Fun ctions into Divergent an d Convergent P ar ts

To explore t he a ctu al r esolut ion of a class of fun ctions int o convergent an d divergent

par ts, we will employ th e L ogarith m ico-E xpon en tial Fu n ct ion s or L -fu n ct ion s first

int roduced by Pau l du Bois-Reymond (10) to measur e the wider

pa nora ma of divergent beha vior left out of Cau chys limit form ula tion. These

fun ctions pr ovide one scale for meas ur ing ra tes of divergence, an d wh ile t hey a re

ultima tely ina dequate to the full task , they cert ainly encompass a wide enough class

of divergent fun ctions to cover m ost of the cases foun d in a na lysis. Accord ing to

G.H.Ha rdy No fun ction ha s yet present ed itself in an alysis the laws of whose

increase, in so far as t hey can be stat ed at all, can not be sta ted, so to say, in

logarithmico-exponential terms.(11)

The L-functions are defined as those functions that result from applying the four

opera tions of arith met ic, th e extra ction of roots, a nd t he a pplicat ion of th e

exponential and logarithmic functions a finite number of times. Du Bois-Reymond

also int roduced the notion of asym ptotic rela tions am ong rea l functions. For

example, we write f > g iff/g tends to infinity and

f g iff/g converges t o 1. With th is nota tion, we ha ve th e following th eorem ,

proved by Har dy (12)

Theorem 2

Any L-fun ction is u ltima tely cont inuous, of const an t sign, an d m onotonic an d, as x

increases, ten ds to infinity or conver ges to a limit. If f an d g ar e L-fun ctions, then

one of th e relat ions f > g , f cg (c const an t) holds between t hem.

The L-fun ctions t her efor form a linearly ordered cont inuu m of fun ctions, with some

convenient properties. Now consider th e previous example of th e ha rm onic series.

Using L-fun ctions a nd t heir relat ions, we can sta te t ha t:




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f (n )=1+ 12

+ 13

+ + 1n -1

log (n ) (12)

which t ells us someth ing useful about t his series, th at it increases at t he sam e rat e

as t he logar ith m, even th ough we do not ha ve a simple closed form expression for t he

series. If we regar d th e L-fun ctions a s a kind of linear scale for m easu ring r at es of

divergence (or convergence) then we can obtain mu ch more inform at ion about th is

series. In fact, th is particular series was studied by Eu ler himself in this light an d

ha s become a s ta nda rd t extbook examp le of asym ptotic an alysis (13). Thus, we have a

succession of stat ement s:

f (n )- log (n ) =0.57721

f (n )- log (n )- -12n

(13)

f (n )- log (n )- + 12n

-1

12n 2

which can be prolonged indefinit ely. This successive an alysis reveals tha t th e

sta nda rd definition of a convergent process is just one st op on a scale of rat es of

increase, an d when we look at th e bigger pictu re, th e distinction m ay not be tha t

importa nt . At a ny ra te, we can consider t he possibility of an alyzing a divergent

function on a scale of specially selected L-fun ctions as a mea ns of isolat ing divergent

and convergent parts, as x ten ds to infinity. Const ru cting a s uit able scale is

somewhat problema tic, due t o the complexity of the L-fun ctions with r egard to th e

relat ions >, an d .

In deed, Cau chys origina l limit definit ion wa s plagu ed by th e complexities of th e rea l

nu mber system , the full ram ificat ions of which wer e not appr eciated a t t he t ime. In

fact, with out a full invest igation of th e logical foun dat ions of the nu mber concept

itself, a t ask which occupied some of th e best mind s of th e 19-th cent ur y, Cauchys

limit opera tion was invalid. We are u p against a similar pr oblem h ere - th e L-fun ctions, and r eal fun ctions in general, ar e even more complex tha n t he r eal

nu mbers . Neverth eless, we can s et up some scales of fun ctions which a re roughly

an alogous t o ra tiona l num bers, and a llow us to obta in ma ny pra ctical results.




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For exam ple, we can form a set of functions, B, by itera ting th e logarith m a nd

exponent ial fun ctions. First we define the iterat ed logarithm k by sett ing

k +1(x )= log {k (x )}an d then define B as th e set of compound fun ctions of th e

form

c exp {cm km (x )exp {cm -1km -1(x ) exp {c0k0(x )} } (14)

where c and t he ciar e non-zero real n um bers. Since this form is compa tible with

th e opera tions used to const ru ct t he L-fun ctions as a wh ole, it defines a subset of

linear ly ordered functions. Specifically f, g B either f


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m (f ) conver ges as x ten ds t o infinity. Thus for finitely redu cible functions we

ha ve a resolution into two par ts, a divergent par t given by

(f )+ ((f ))+ (2(f ))+ + ( m -1(f )) (16)

an d a convergent pa rt given by m (f ) .

5. The Gen er alized Lim it

Now that we have a mean s of separa ting at least one class of divergent functions int o

divergent an d convergent pa rt s, we can intr oduce a genera lized limit operat ion by

defining th e limit of an y f in F simply as t he limit of its convergent p ar t. Since th is

new limit opera tion depends on the set B which we used to effect a separ at ion int oconvergent a nd divergent pa rt s, we will write

l imx

{f (x )}(B )= l (17)

Such a generalized limit is a tr ue extension of the st an dar d limit defined by Cauchy,

since the convergent par t of a convergent function is just th e fun ction itself.

The abstr act tra in of th ought th at ha s lead u s to this new limit opera tion m ay seem

obscur e. One might well ask if th is genera lized limit rea lly makes a ny sense in

ter ms of th e new valu es it int roduces int o an alysis. Per ha ps it is highly ar tificial ,

producing results t ha t would be of little pra ctical use. And its r at her abstr use form

ma y ma ke it difficult to apply in pr actice as a problem s olving tool.

Of cour se, ther e is no way to an swer su ch legitima te concerns with in th e realm of

logic. But we can pr esent some specific resu lts based on th e gener alized limit th at

ma y help the rea der an swer some of th ese quest ions. Takin g initially th e case of

discret e infinit e processes, wher e th e varia blex ta kes positive integer values , we

can obta in t he following resu lts




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1r

r =1

= l imn

1r

r =1

n

(B ) = (18)

rr =1

= limn

rr =1

n

(B ) =-0.2078865 (19)

1u + r

r =1

= limn

1u + r

r =1

n

(B ) =- ' (u )

(u ) (20)

log (r )

r =1

= limn

log (r )

r =1

n

(B ) =1

2log (2) (21)

2r

r =0

= l imn

2r

r =0

n -1

(B ) = -1 (22)

Man y similar examples could be added t o th is list. If we compa re (20) to a st an dar d

convergent series for th e logarith mic derivat ive of th e gamm a fun ction, n am ely:

- + 1r

- 1u -r

r =1

= ' (u )

(u ) (23)

we find th at th e gener alized limit a pproach has simplified the form ula. The series

(20), although simpler a nd m ore direct th an (23) is unfort un at ely regarded a s ha ving

no nu mer ical mea nin g. Or in t he case of (22), which, like th e divergent series (7) is

regarded a s ha ving at best a pur ely form al mean ing, we find th at it repr esents a

convergent process with a n um erically soun d value.

Let us der ive a value for t his series in both ways. First , if we set

V = 1 + 2 + 4 + 8 + (24)




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an d th en m ultiply th rough by 2, pretending th at th e series is absolutely convergent ,

we find th at

2V = 2 + 4 + 8 + 16+ = V -1 (25)

Fr om which we deduce tha t V = -1. On th e oth er ha nd, set

f(n )=1+ 2+ 4+ + 2 n -1 (26)

We can show that l imn

{f (n )}(B )=-1. Start ing with th e identity

1+ 2+ 4+ + 2 n -1=2 n -1 (27)

we see tha t fcan be measured with a sta ndar d member ofB

(f(n ))=2n = exp {(log 2)n } (28)

an d we can th erefore obtain a value for :

(f ) = f - (f ) =2n -1 - 2n = -1 (29)

Since this is a const an t, f is finitely reducible and converges in our more general

sense t o th e value obta ined by form al ma nipu lation of th e series.

In pra ctice, it is not necessary to repr oduce the full argum ent based on and

opera tions. If it is possible t o find a linear combina tion of fun ctions from th e set (B)

th at , when subt ra cted from a divergent function t ur ns it int o a convergent pr ocess,

th en we know th at we have a genera lized limit. This means th at we can work with

finitely reducible processes almost as th ough th ey were convergent in t he u sua l

sense. Tha t will allow us t o ta p th e considera ble body of techniques th at ha ve beendevised over the past two centuries to manipulate limit operations.

This techn ique also mean s th at we can often r eform ulat e known results to derive




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generalized limits. For example, consider a st an dar d asympt otic resu lt used in th e

derivat ion of St erlings appr oximat ion t o th e factorial:

log (r )r =1

n

=(n + 12

)log (n ) - n + log 2 + on (30)

where on ten ds to zero as n t ends to infinity. Since the divergent t erms on th e right

ar e members of (B) , th is equa tion can be rewritt en as

l imn

log (r )r =1

n

(B )=

= l imn

log (r )-(n + 12

)log (n ) + nr =1

n

= log 2 (31)

Thus finite reducibility simply means th at th e divergent pr ocess can be rewritten as

a convent iona l limit.

We can also trea t m an y of th e fun ctions of class ical an alysis by int roducing

additional parameters into the functions of the set B. This can be accomplished by

allowing th e const an ts in (14) to depend on th ese additiona l par am eter s. With th is

convent ion, we can obta in resu lts su ch a s

(u )= l imn

1r u

r =1

n

(B ) (33)

11 - u

= limn

u rr =0

n -1

(B ) (34)

While fam iliar in form , it is importan t t o realize that th ese stat ements hold for a ll

rea l values ofu . With t he convent iona l limit operat or, convergen ce break s down for

u < 1 in th e zeta function ser ies, an d for | u | > 1 in t he series for 1/(1-u ) .




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6. The Most Gener al Fra mew ork

We have so far demonst ra ted our t hesis, tha t divergent pr ocesses can be resolved into

un ique divergent an d convergent pa rt s, on a rest ricted class of fun ctions. Indeed,th ere ar e L-functions wh ich a re n ot equivalent, in t he a symptotic sense, to any

mem ber of (B ) and even if we ha d a complete ba sis set for t he L-fun ctions, ther e ar e

still ma ny additional fun ctions left out . For exam ple, sin (x ) is not asymptotically

equivalent to an y L-fun ction becau se it is boun ded a nd oscillates in definitely.

And, if we look more closely at (7), we will find t ha t it is not finit ely redu cible eith er.

Borr owing from Ph ipps an alysis of th is series we know th at

1+ 1x + 2x 2+ 6x 3+ + (n -1)!x n -1 (34)

is repr esent ed by an a sympt otic series of the form

n !x n c1(x )

n+

n !x n c2(x )

n 2+

n !x n c3(x )

n 3+ (35)

The variable x actua lly represent s a par am eter, which we consider t o be a fixed

quan tity as the active variable n ten ds to infinity, and t he functions ci (x ) ar e

defined by recusion on t he in dexj. This expr ession is complicated by t he factorial

function, which we here ta ke to be the discret e version of th e gam ma fun ction. Thegamm a function is not a m ember ofB, but even if it were, we would st ill ha ve a

situ at ion wher e the process is not redu cible in a finite nu mber of steps t o a

convergent function. For an y positive int eger m we would h ave

m 1+ 1x + 2x 2+ 6x 3+ + (n -1)!x n -1 n !x n cm (x )

n m (36)

which will not t end t o zero as n ten ds to infinity no ma tt er how lar ge m .

Thus we appea r t o be at a n impa sse and we might conclude th at , while gains can bema de in gener alizing limits t o pat ch some of th e dam age done by divergence in

specific cas es, Ph ipps cont ent ion t ha t divergen ce can be era dicat ed from a na lysis

oversta tes t he case. But t his problem, along with th e problem of th e incompleten ess




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of th e L-fun ctions can be solved - we only n eed t o gener alize our forma lism.

The limit is s ta ted in ter ms of the beh avior of specific fun ctions, but it can also beviewed as a sta tement about m embership in sets of functions. The stat ement t ha t

l imx

f (x )= l is equivalent to saying th at

f g | o O , g (x )= l + o (x ) (37)

where O repr esent s th e set of fun ctions which t end t o zero as x ten ds to infinity, and

l sta nds for t he function wh ich is equal to the const an t value l for all x . If we

adopt th e convent ion t ha t a n operat ion between set s of fun ctions denotes t he set

obta ined by perform ing th e opera tion between a ll pairs of elements ( i.e.S +T = u | u = s + t , s S and t T ) then (37) can be writt en m ore su ccinctly as:

f l + O . The relations


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[ f ]=[(f )]=(f ){1+ O }=(f )+ (40)

an d since f [f ] this shows tha t

f (f )+ (41)

We also ha ve the mem bership relat ion

f (f )+ [f -(f )]=(f )+ [(f -(f ))]

=(f )+ (f -(f ))+ (42)

If we continu e th is process to n + 1 steps, we obta in:

f (f )+ ((f ))+ (2(f ))+

+(n (f ))+

= (f )+ ((f ))+ (2(f )) +

+(n -1(f ))+ [(n (f ))]

= (f )+ ((f ))+ (2(f ))+

+(n -1(f ))+ [n (f )] (43)

which is th e set version of

f =(f )+ ((f ))+ (2(f )) + +(n -1(f ))+ n (f ) (44)

Now suppose that

n -1

(f ) diverges bu t

n

(f ) converges, th en t he set version,equat ion (43) expresses t he fact th at fcan be resolved into a divergent par t, th e finite

s u m




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(f )+ ((f ))+ (2(f ))+ (3(f )) + +(n -1(f )) (45)

and a convergent part which is here an equivalence class of functions, all of which

mu st converge to the sam e limit.

Thus , fcan be cha ra cter ized by two descendin g sequences

(f )>>((f ))>>(2(f ))>> >>(n (f ))>> (46)

a n d

(47)

Both sequences are defined in t erm s of th e opera tors and which can only iterat e a

finit e nu mber of tim es. But t he second r epresent at ion is not limited t o finite

operat ions becau se we can t ake th e inter section of an infinite nu mber of sets and

obta in a well defined set . Cons ider th e following ana logy with r eal nu mber s: sup pose

that x 0> x 1> x 2> , x n >0 is a decreasing sequen ce of rea l nu mbers an d define

the set S as

S = [0, xn )n =0

(48)

wh er e [ 0, a) represent s th e set of nu mbers less than a and greater tha n or equal to

0. We can prove tha t S is also an interval of th is type,which m eans t ha t t here is some

number l such tha t S = [0 , l) an d, in fact, l is th e limit of th e sequence.

Suppose, for th e sake of ar gument , tha t n (f ) diverges for a ll positive int egers n ,

but when we form th e intersection

S =

n =0

(49)

we find th at S is a set of the form < g> where g is a convergent fun ction. In a na logy




23/26


with t he finite case, we can regard th e equivalence class [g] as representing th e

convergent par t of the function f , leading to th e genera lized limit definition

l imx { f (x )}(B )= l imx g (x ) (50)

Anoth er a dvan ta ge of set t heory is the possibility of cont inu ing th is process furt her .

Suppose that g tu rn s out t o be divergent . We can st ill cont inue ta king finite

iterations n (g ) un til we ar rive at a convergen t fun ction or else form a second

infinite intersection

S (2) = n =0

(51)

an d so on. All of th is can be neatly summ ar ized in a definition u sing tra nsfinite

ordinal nu mbers :

Definition 1 (Tra ns finit e Redu ctions )

Let denote an ordinal nu mber. The operat or (f ) is defined as follows

(1) 0(f )=(f )

If has an immediate predecessor then

(2) (f )= ' (f )-( ' (f ))

and if ha s no immediat e predecessor t hen

(3) (f )= suprem um of <




24/26


By using ordina l nu mber s, we can give th e resolut ion of a fun ction int o convergent

an d divergent par ts t he widest possible lat itude. Of cour se all of th is is dependent onhow we choose a representative function from each equivalence class, and at this

point we do not h ave an y gener al t heorems on h ow th at choice will affect t he

definition of a genera lized limit. But t his approach does assur e us t ha t, however t ha t

choice is made, the resu lting theory has a logical basis and t ha t is th e primar y

objective of th is first at tem pt at era dicat ing divergence from a na lysis.

We should also ment ion a t t his point t ha t t his form alism leaves out an import an t

class of fun ctions: th ose th at ar e boun ded but n ot convergent . In fact it is easy to

show t ha t for a ll choices of basis functions (f ) conver ges t o zero wh en fis

boun ded, so th at th e reduction operat ion does not distingu ish between convergent

fun ctions a nd divergent , but boun ded, fun ctions. Thus in th e most genera l case we

ma y only find th at (f ) is boun ded as x tends to infinity.

Interestingly, it is possible to prove that for any choice of basis functions, all bounded

fun ctions a re finitely reducible. This fact resu lts from t he fundam enta l theorem th at

every vector spa ce ha s a basis. Since the s et of all boun ded functions is closed u nder

th e sum of two fun ctions a nd t he pr oduct of a function an d a const an t, it is a vector

space. The essent ial new step h ere is to const ru ct a vector spa ce based on th e

behavior of functions as x tends to infinity. From tha t we can deduce th at a nyboun ded fun ction is a finite linear combinat ion of boun ded basis functions. If we

include the function f(x ) = 1 in th e basis set, t hen its coefficient can be ta ken a s t he

limit of a bounded function.

The pr oblem of extr acting a limit from bounded divergent functions is a distin ct

problem t ha t n eeds to be studied in t he cont ext of oscillat ory operat ions su ch a s

Fourier Tr an sform s an d Fourier Series, for wh ich t here is a large body of results.

This type of problem does not a rise with L-fun ctions becau se th ey are u ltima tely

monotonic, and a boun ded monotonic function always conver ges to a limit. We will

conclude by defining wh at we mea n by a resolution of an y L-function in to divergentan d convergent pa rt s. While not complete, th is definit ion repr esent s a big step in th e

direction we wan t t o tr avel, towar d a complet e an d divergence free calculu s.




25/26


Definition 2

Suppose that f is an L-fun ction, and a choice operat or t ha t a ssigns a n

asym ptotically un ique represen ta tive to every L-function. If for some ordin al nu mber

, (f ) is convergent while (f ) is unbounded when


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(9) T.E.Phipps, Jr. , Heretical Verities: Ma th ema tical T hem es in Physical

Description (Classic Non-Fiction Libra ry,Ur ban a Il.,1987) , Cha p. 12

(10) G.H.Har dy, Orders of Infin ity, (Hafner , NY, 1971)

(11) Ibid, 35

(12) Ibid, 18

(13) K. Knopp, T heory and Application of Infin ite S eries , (Dover , NY, 1940), P.527


i d i f i i ll bli h d i h 5

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A New Perspective on the Foundations of Analysis