Advice
We advocate moving from "functions" as the central objects of analysis to "formal power series" or "polyseries". In some sense this goes back to Euler and even Newton, and provides a powerful combinatorial alternative requiring no waffling about "evaluations" at "real numbers" that cannot in fact be explicitly computed. The painful reality is that the current over-reliance on "transcendental functions" is largely wishful thinking, obviously unsupported by the computational capacity of our machines.
A central object in this new story, involving formal power series/polyseries, are the exp and log polyseries, and here we introduce grounded variants called E and L, which are miraculously inverses with respect to composition. Demonstrating this involves the formal Faulhaber Derivative D that plays such a big role in the Algebraic Calculus One course.
In fact this material is very much in the direction of the Algebraic Calculus Two course, which is under preparation.
This lecture is part of the Exploring Research Level Maths series of videos, available to Members of the Wild Egg Channel, which includes Playlists on Dynamics on Graphs, The Hexagrammum Mysticum, Algebraic Calculus and Curves and general Advice. In the Advice Playlist, we are currently looking at Orthogonal Polynomials in a novel two dimensional number theoretical way using maxel theory. This video is part of that development.
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0:01
hello everyone i'm Norman Wildberger thanks for joining me on this adventure we are giving you
0:06
some advice and i want to make contact with important ideas from calculus today and shift
0:13
your understanding of functions towards formal power series which are sort of combinatorial
0:20
versions of of power series this is a very important shift because from a logical point of
0:26
view i'm going to argue the current understanding of functions as as things that input real numbers
0:33
and output real numbers really doesn't work computationally okay but there is an alternative
0:37
that the combinatorialists have been building up for uh you know actually quite quite some time
0:43
and and this is really the the stronger and more solid foundation for a lot of analysis
0:50
and we're going to look very explicitly at uh two of the most important and central uh power series
0:58
the exponential and the logarithm poly series but i'm going to give them to you in terms of two
1:04
variants called e and l which are going to fit more naturally with our our understanding of the
1:11
the composition operation on poly series okay so let's start with formal power series okay this
1:19
is a a concept which probably was officially introduced first by ernst schroeder in a paper
1:26
uber iterte function in math annals or analyn perhaps in 1871 and then considerably later
1:38
ivan niven wrote a paper called formal power series in the american math
1:43
monthly in 1969 which was a quite a celebrated paper and and really i think brought this to the
1:51
wide attention of the mathematical community steven scheinberg also wrote a paper power
1:55
series in one variable the following year which also sort of gave an overview of things
2:00
and in terms of sort of a comprehensive treatment you can't really go past richard stanley's
2:05
monumental work enumerative combinatorics in two volumes which was published in 1997 and 1999
2:15
and there's quite a lot of information about formal power series there so the basic idea
2:21
is to look at power series not as functions but as sort of formal polynomials that just keep on going
2:31
or we call them poly series and in fact this point of view is is really uh historically
2:38
going back to oil or at least you know who definitely thought in this direction
2:43
and no no doubt almost really to newton who appreciated that
2:47
that power series could be thought of as sort of algebraic equivalence of infinite decibels
2:52
so um i want to reorient you since i'm trying to give you advice i think it's
2:58
really important that you let go of some of the programming that you will have been subjected to
3:04
as modern students of mathematics you've been oriented to think about the function concept as
3:10
sort of all pervasive you know in any situation we look around to define a function we think of
3:17
a function something that inputs one thing and outputs something else it's a dynamical thing okay
3:23
and of course the the standard repertoire of analysis is full of functions
3:28
elementary ones uh you know like f of x equals x squared not so elementary ones like sine of x log
3:35
of x tan of x arc sine of x the zeta function the the gamma function et cetera et cetera
3:42
but if we look carefully at these things okay we see that there's actually a serious problem
3:48
the serious problem is that we're not actually able to evaluate these things with the exception
3:53
of f of x equals x squared that one we can actually evaluate if i ask what's f of 7
3:59
and 7 squared i can calculate that as 49. if i ask you what is sine of 7 or
4:05
tan of 7 or gamma of several gamma 7 we can do a gamma of say 7 over 3 or zeta of 7 you know or
4:16
most of the functions in in modern analysis cannot actually be calculated exactly approximately yes
4:24
but not exactly and really from a logical point of view that really excludes them from being properly
4:31
defined mathematical objects they're really not they are objects from applied mathematics
4:39
that we import into pure mathematics and sort of pretend that they are now correct
4:45
precise pure mathematical objects when they are not okay so i want to suggest to you politely
4:54
that it's not a question of opinion here it's not that oh well we could take this point of view and
4:59
we could or we could take this point of view you know we could either think about these things as
5:02
functions or as as as power series formal policies no okay the the functional approach is logically
5:10
weak it's logically compromised the formal power series approach of the combinatorialists
5:19
gives us an alternative more solid foundation okay so having said that and this is you know
5:27
something that i'm going to have to repeat over and over again because our programming is just
5:30
so all pervasive but anyway let's look at two of the central examples okay the exponential and the
5:38
logarithm poly series and i'm going to introduce some variants called capital e and capital l and
5:46
you'll see why um and they this fits very nicely into this framework of composition of poly series
5:57
so let me remind you that if we have two poly numbers p and q so these are like finite
6:01
polynomials the composition of p and q is sometimes denoted p circle q okay which is
6:08
another way of saying p of q and that this kind of composition extends to poly series where the terms
6:16
are ongoing provided that the q inside here is grounded in other words that it can be written as
6:23
a power series or poly series an alpha with no constant term okay that's the
6:29
crucial property that allows such a composition to be in general defined
6:36
and then once we have this notion we can say that a polyserious p has a compositional inverse q
6:41
precisely when p circle q equals q circle p equals the compositional identity which is just a
6:48
polyserious alpha and then i mentioned a theorem like this last time that you know poly series
6:54
uh p c zero plus c one alpha plus c two alpha squared etc has a compositional inverse precisely
7:00
when there's no constant term when c zero equals zero and when this c one term the coefficient of
7:07
alpha is non-zero and in this case the inverse q is unique and can be computed sort of term by term
7:16
so in particular a poly series must be grounded in order for it to have a compositional inverse
7:21
right the grounded condition is that c zero equals zero but a little more is actually required we
7:27
also need that this c1 is not zero okay so now a central object the exponential poly series x
7:38
okay here it is it's a poly series that's written in summation notation like this summation k equals
7:44
0 and then onwards going through natural numbers proceeding further and further 1 over k factorial
7:52
times alpha to the k and then explicitly the first few terms 1 plus alpha plus 1 half alpha
7:56
squared plus 1 6 alpha cubed et cetera we have the various factorials appearing here
8:02
now this is not grounded right it's not grounded because there's a constant term which is one
8:07
so if we want to move into a domain where we can talk about
8:12
compositional inverses we're going to have to modify this by stripping away the constant
8:18
term so we're going to do that and we're going to give that new thing this sort of grounded version
8:24
of the exponential poly series a new name call it capital e so here's capital e which is x minus one
8:32
we'll call that the grounded exponential poly series here it is right here okay it looks just
8:37
like the exponential except that it starts with alpha alpha plus one half alpha squared and so on
8:47
this is further unital uh maybe i should put that in bold so until is referring to the fact that
8:55
this term alpha actually has a coefficient of one it's not just non-zero it actually is equal to one
9:02
that's particularly pleasant okay and so now we come to uh one of the great miracles of calculus
9:11
uh just a a beautiful fact that uh you know has manifestations throughout the subject
9:19
and i want to emphasize that this in the context now of this framework of power series or
9:27
poly series okay the compositional inverse of e
9:31
this is now a poly series which satisfies our condition so we know that it has
9:38
a compositional inverse and now we're going to say what that compositional inverse is
9:42
the compositional inverse of e is what we're going to call the grounded logarithm poly series
9:48
and we'll call it capital l and here's what it is it's this series here alpha minus one half alpha
9:56
squared plus one third alpha cubed minus one quarter after the fourth plus etc alternating
10:03
and sine the denominators here are growing much slower than over here here they were factorials
10:09
here they are just natural numbers by themselves okay so in summation notation summation k
10:15
equals 1 onwards minus one to the k plus one over k alpha to the k so when k is one uh we get alpha
10:24
we get a one in the denominator and we get um minus one squared so we get a positive term there
10:30
okay so in calculus courses you usually don't see this thing defined usually
10:38
there's another so-called function called the log function written either with you know the symbol
10:44
log or sometimes la lawn with an ln and what we're expressing as l would be in a traditional calculus
10:52
course expressed as log of one plus x or or lon of one plus x the actual lawn or log function
11:01
in function in quotes um in calculus is is not typically defined in terms of a power series
11:08
um because that's actually problematic but it's defined in some other fashion um often
11:14
as an integral or sometimes as the inverse of of x okay but what we want to do is we want to separate
11:22
ourselves from any understanding of these things as s functions okay so we're going to uh stick
11:28
with the purely algebraic approach here functions are not required and so the the definition of l or
11:35
the way this l comes about is we're looking for the compositional inverse of this exponential e
11:44
okay so let's look at some properties of e and l okay so
11:51
these these two power series have lots of remarkable important properties and when
11:58
we want to express some of them it's convenient to move to bi-poly numbers or by poly series where we
12:08
think not just in terms of alpha but also beta so you make recall that the alpha poly number is the
12:17
vertical list 0 1. well there's also a horizontal list 0 1 and we call that beta and that way we
12:25
can think about uh sort of bipoly numbers poly polynomials in in two variables alpha and beta if
12:32
you like in terms of arrays two dimensional arrays of coefficients instead of just one dimensional
12:39
lists okay so in the in this context of sort of expanding our arithmetic to include
12:46
both alpha and beta then we can express this very important property of the grounded exponential
12:53
poly series e this way that e of alpha plus beta equals e of alpha plus e of beta plus e of alpha
13:03
times e of beta and so before i go on to the graph the grounded logarithm i think it's
13:11
um good for me to try to justify this so let's let's see if we can sort of you know um convince
13:16
ourselves of this all right so let's define first of all e say alpha and i'll just grab the first
13:26
a few terms of this maybe up to up to degree six let's say
13:32
all right up to there and we'll make that definition and now let's look at e of
13:39
alpha plus beta okay what is that so there it is there let's expand this out
13:48
and let's now sort of cut this down so that we're only going
13:55
well i guess we're only going up to degree uh six aren't we going up to degree six okay fine and
14:07
that was the left hand side and now let's look at
14:11
the right hand side e alpha plus e beta plus e alpha e beta let's see what that looks like
14:18
there's alpha there's beta there's the product let's expand that
14:27
so we've expanded it up to here
14:32
and um okay how does this compare with this
14:41
well i am
14:47
if if we sort of cut down things in the first expression i only went up to a total degree of
14:52
six okay so if i if i only go up to total degree of six i am going to um kill these
14:58
guys these are all a degree more than six and the other ones um yes uh this one here is more
15:04
i'm going to just go up to degree six and no further uh to die anymore this is bigger bigger
15:17
these terms are also bigger
15:23
that one's bigger
15:27
how does that look go ahead just terms up to the total degree six
15:34
okay so um let's let me just expand that out so that's what it is and
15:41
up here what i had originally let me just
15:45
look at that okay so i'm comparing uh this with this are these all the same i
15:53
started to remove some some further ones i forgot these are all bigger degrees get rid of them um
16:05
okay i think i think i've done that now so now if you if you compare this thing here i've gone to
16:10
up to degree uh six total degree six and this expression here total degree six and it looks
16:17
like these uh these things are are the same right this thing here is the same as this so we'll see
16:23
these two expressions these two expressions agree up to total degree six in alpha and beta
16:38
okay so that convinces us at least that this is very likely a
16:42
true expression it's not a proof of course but it's on the way of a verification so that was
16:48
for this grounded exponential poly series now what about the grounded logarithm poly series
16:56
well here is the uh the crucial relation for it which is very analogous to the one for the
17:04
grounded exponential it's that l of alpha plus beta plus alpha beta equals l alpha plus l beta
17:11
in fact i might just juxtapose these two things so
17:19
so just put that there to remind us so that's the relation for the
17:25
grounded exponential here's the relation for the grounded logarithm l of alpha plus beta
17:30
plus alpha beta equals l alpha plus l beta and these two relations are of course intimately
17:36
connected with each other because these two poly series are compositional inverses and i remind
17:42
you what that means is that if we take e of l we get alpha and if we take l of e we get alpha
17:55
so how do we prove relations like this well it turns out that there's a highly non-obvious tool
18:02
here that people have realized that one can use to establish all kinds of interesting relations
18:08
here it's a very general kind of tool it's the foul hyperderivative d actually i might call
18:14
it file hover capital d derivative that's what i typically do in the algebraic calculus course so
18:20
this of course plays a major role in the algebraic calculus and we are emphasizing that this is a
18:26
formal operation on polynomials or poly numbers or poly series it's defined just in the simple way
18:35
that d of a pure power alpha to the n is n times alpha to the n minus 1 for any natural number
18:42
at n equals 0 1 2 3 et cetera with the understanding that when n is 0 in other
18:47
words d of a constant or d of 1 we get 0. and then this is extended linearly so if we want to take d
18:55
of this poly number 2 plus alpha minus 7 alpha cubed plus alpha 5 well in d of 2 which is 0 d
19:02
of alpha which is 1 d of minus 7 alpha cubed which is 21 alpha squared and d of alpha the fifth which
19:09
is five out of the fourth and then it's elementary to extend this to polyseries you just apply the d
19:16
to each of the individual powers in a poly series so then there are two important rules uh that this
19:24
formal final hopper derivative satisfies and and i i really want to emphasize that this is um
19:31
quite logically independent from the usual understanding of a derivative as a slope of
19:35
a tangent you know to a function we don't have functions here we just have these poly series
19:40
and it's also independent of any kind of uh limit of of ratios like f of x plus h minus f of x or
19:48
over h you know some limit of slopes of secants or some such thing okay this is a purely algebraic
19:55
thing does not require any infinite processes and it also does actually not really require a prior
20:02
understanding of so-called functions we're just operating this purely algebraic world
20:08
okay so here are the uh two crucial um sort of rules the the product rule for the derivative d
20:15
of p times q is p times d of q plus q times d of p and the derivative of a a composition so
20:26
called chain rule d of p circle q equals d of p circle q that's one poly number times d of q
20:38
and these formulas also extend to poly series when they are defined in other words when these
20:43
compositions for example are working okay which would require q to be a a grounded poly series
20:53
okay so now we can go up to the um the l and the e poly series and we can calculate their uh
21:02
formal derivatives so d of l what is that well let's uh let's go and pull these things uh
21:09
again for us so that we can remind ourselves so let's grab this e and that l
21:18
come down here okay so i'll just stick that over here so there's that one there
21:25
and then there's there's that one so here's the definition of e and here's the definition of l
21:29
so if we take the derivative of l that means we're just taking the derivative of each individual term
21:34
the derivative of alpha is one the derivative of minus one half alpha squared well the two comes
21:40
to the front cancels with the two that's there and we just get alpha and then the next term
21:44
derivative of one third alpha cubed is plus alpha squared the derivative of minus one quarter alpha
21:50
to the fourth is plus one quarter et cetera so we get a series which has a natural interpretation as
21:56
one over one plus alpha that's a rational function okay that's like a geometric series that
22:04
algebraically is equivalent to one over one plus alpha so the derivative of l is a very pleasant
22:13
thing again this is a very important relation what about the derivative of e well you'll probably all
22:18
know that that the derivative of e is more or less itself but remember this is not quite the
22:22
exponential because we stripped away the one so we take the derivative of this thing when you get one
22:28
plus the derivative this thing is alpha the derivative of this thing is alpha squared but
22:32
there's a two in the bottom okay the derivative of this one the four comes out cancels with the
22:37
the four and the four factorial here and we just get 1 over 3 factorial so what we end
22:42
up getting is the usual exponential poly series or in terms of capital e it's 1 plus capital e
22:51
all right so now let's put these things together to try to understand
22:57
what happens when we compose l and e so let's do it in this order so let's apply d
23:02
to this composition l of e in other words l circle e so d of l circle e
23:08
according to this chain rule that's going to be d of l circle e times d of e well d of
23:18
l we've calculated that's 1 over 1 plus alpha so we can put 1 over 1 plus alpha there
23:26
and we are taking the circle of that we're composing that with e
23:31
what does that mean well it just means that we're replacing the alpha
23:35
with an e and what is d of e well dv is one plus e so we're getting one plus e times
23:42
this thing evaluated at e is one over one plus e so we get one plus e over one plus c which is one
23:51
so the derivative of this poly series l circle e whatever it is is just the constant one
24:00
well that means that's a very strong condition that means that this l circle e has to only
24:07
consist of constant and linear terms in fact the linear term has to be alpha because the derivative
24:13
is going to be one however we don't know what the constant term is because when we take the
24:18
derivative that's kill that's sent to 0 no matter what it is so to evaluate what this c0 is we can
24:27
plug in a simple value we can plug in what happens if we replace alpha with zero we can evaluate left
24:34
and right hand sides at zero so l of e of zero well e is zero here's e if you plug in zero you're
24:43
gonna get zero and so you're gonna get l of 0 and when you plug in 0 for l you're going to get 0.
24:49
so you're going to get 0 on the left hand side the alpha is 0 and that proves that c 0 equals
24:58
zero and so that there's no constant term and so l circle e is actually alpha so we've proven this
25:07
strong compositional connection between e and l in this sort of magical way by utilizing this formal
25:15
derivative it's a very powerful technique so now we have uh two central objects of calculus and uh
25:23
and now the kinds of considerations that we've been investigating now expand much further