# Algebra of divergent integrals

• A
Anixx
TL;DR Summary
Call to discuss an extension of real numbers that includes divergent integrals and series
Hello, guys!

I would like to know your opinion and discuss this extension of real numbers:
https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651

In essence, it extends real numbers with entities that correspond to divergent integrals and series.
By adding the rules derived from Faulhaber's formula, it allows some wonderful things to be done,
such as expressing a derivative of an analytic function without using limits or infinitesimals, and also expressing trigonometric functions via inverse trigonometric in closed form.

Some physical expressions, especially those derived via regularization become simplier.
For instance, the mean energy of quantum harmonic oscillator can be written instead of
$$\varepsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /kT}-1}}$$
as
$$\varepsilon =h\nu\omega_++kT e^{\frac{h\nu\omega_-}{kT}}$$
with underlying implication that the full energy (before regularization) is infinite.

Your thoughts? I mostly interested in the opinion of
1 Algebraists - on what algebraic properties such extension should have
2 Physicists - on physical applications in the areas that require regularization

At face value, this question asks how to add certain strings of symbols to the real number system. If we view a divergent integral as a string of symbols then immediately we have questions like how to decide if two different strings of symbols represent the same divergent integral. For example is ##\int_0^\infty x\ dx## to be the "same" divergent integral as ##\int_0^\infty (1+x)dx##?

So I don't see that inventing an algebra for divergent integrals can proceed until the subject of divergent itegrals is treated at the level of a formal language. For example, what are the rules for creating strings of symbols that are "well formed formulas" in the language of divergent integrals. What are the rules that define an equivalence relation between two different strings of symbols? - i.e. how do we know that two different strings of symbols are to be treated as the same divergent integral?

Anixx
Well, from the axioms listed in the link, $$\int_0^\infty (1+x)dx=\int_0^\infty xdx+\int_0^\infty dx$$, so they are not equal. The last term even has special symbol.

Well, from the axioms listed in the link,

Which axioms?

The axioms stated in the link do not say:
If A,B,C are each symbolic expressions for a extended integral and the symbolic expression formed by "A = B + C" is evaluated as true then the symbolic expession "A = B" must be evaluated as false.

In fact, the axioms in the link are not formulated to deal with establishing a formal language.

Anixx
The axioms on the link establish linearity:

$$\int_a^b f(x)+g(x)dx=\int_a^b f(x)dx+\int_a^bg(x)dx$$

So, iff

$$\int_a^b f(x)+g(x)dx=\int_a^b f(x)dx$$

then $$\int_a^bg(x)dx=0$$

But $$\int_0^\infty dx$$ cannot be equal to 0 because it is not regularizable by a stable method (Cesaro, Abel, etc). It diverges to infinity, which excludes it being 0.

So, iff

$$\int_a^b f(x)+g(x)dx=\int_a^b f(x)dx$$

then $$\int_a^bg(x)dx=0$$

That conclusion must be stated as an axiom. The deduction you made is correct if applied to integrals that have values defined in the ordinary way. Integrals not defined that way are merely strings of symbols.

But $$\int_0^\infty dx$$ cannot be equal to 0 because it is not regularizable by a stable method (Cesaro, Abel, etc). It diverges to infinity, which excludes it being 0.

Where is this stated in the axioms? You seem to be saying that the notation "##\int_{0}^\infty dx ##" represents something that has defined properties. What definitions define the properties of the thing represented by this notation? You say it "diverges to infinity". Are you saying the notation "##\int_{0}^\infty dx ##" represents an infinite sequence? What sequence does it represent?

Anixx
Well, if we have two elements "zeroes", such that
$$A+0_1=A$$
and
$$A+0_2=A$$

then

$$0_1=0_1+0_2=0_2+0_1=0_2$$

So, the both zeroes are equal. This comes from commupativity of addition, which is true for any vector space. Maybe I should state that this space is a vector space and even a ring, but this is too trivial (as we are talking about extension of real numbers).

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Anixx
Are you saying the notation "" represents an infinite sequence?
Well, since we are talking about integrals rather than series, $$\int_0^\infty 1 dx$$ represents the behavior of function $$f(x)=x$$ at positive infinity (one may call it germ). And yes, it can be represented as a series or a sequence as well:
$$\int_0^\infty 1 dx=1/2+\sum_{k=1}^\infty 1=-1/2+\sum_{k=0}^\infty 1$$
This follows from the axioms.

After generalizing Laplace transform to divergent integrals, we also will see that it is equal:

$$\int_0^\infty dx=\int_0^\infty\frac1{x^2}dx$$

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Maybe I should state that this space is a vector space and even a ring, but this is too trivial (as we are talking about extension of real numbers).
My understanding is that your goal to find a way of defining operations on some set of strings of symbols in such a way that the set is a field and the real numbers are isomorphic to a subset of this set. If we assume that your have already attained this goal, then, yes, we may assume the properties of a field apply to the set. However, you haven't defined the elements of the set or what the field operations are.

And yes, it can be represented as a series or a sequence as well:
$$=1/2+\sum_{k=1}^\infty 1=-1/2+\sum_{k=0}^\infty 1$$
This follows from the axioms.

We need a definition of what the notation "##\int_0^\infty 1 dx##" means before we can apply any axiom to it.

Are you defining the notation "##\int_0^\infty 1 dx##" to be the infinite sequence whose ##j##th term is ##1/2 + \sum_{k=1}^j 1##?

Anixx
This notation means a divergent integral. Or, should I say, the behavior of the antiderivative of the function under integral sign at positive infinity. The axioms provide criteria of when two divergent integrals are equal.
I do not understand why you consider the behavior of a sequence at infinity as a good "definition" while the behavior of a continuous function at infinity is not a satisfactory definition.

Anixx
The system is definitely not a field as there are non-zero elements by which we cannot divide (at least in the version of the theory referenced here, since I am thinking about a modified version).

This notation means a divergent integral. Or, should I say, the behavior of the antiderivative of the function under integral sign at positive infinity.

But what is the precise mathematical definition of a "behavior"? I agree that "the behavior of the antiderivative of a function at positive infinity" conveys some intuitive ideas. However, to formulate a rigorous mathematical theory, we need more than intuitive notions.

In defining the (ordinary) concept of the integral of a real valued function the notation "##\int_a^b f(x) dx##" denotes a real number if the integral exists. So provided the integrals exist there is no ambiguity in notation like ##\int_a^b f(x) dx + \int_c^d g(x) dx## because this is notation for the addition of two numbers.

If you want to extend the notation so "##\int_a^b f(x) dx ##" represents something other than a number, you must say precisely what this something is. For example, if that notation can now represent a "behavior" then you must explain what the symbol "+" means in that context. What does it mean to add behaviors?

Anixx
As you can see, the axioms postulate linearity: $$\int_a^b f(x)dx+\int_a^b g(x)dx=\int_a^b (f(x)+g(x))dx$$

In other words, a sum of two (divergent) integrals is a (divergent) integral of the sum.

In other words, a sum of two divergent integrals is a divergent integral of the sum.

That only states a property of divergent integrals without first defining what a divergent integral is. It doesn't define what "+" means in the context of divergent integrals. If a divergent integral does not represent a number then we have no guide as to how to interpret "+" , because the "+" can no longer be assumed to represent the addition of numbers.

Anixx
The above formula is the definition of addition. And, by the way, you can see that from this axiom follows commutativity of addition, which in turn proves uniqueness of zero.

The above formula is the definition of addition. And, by the way, you can see that from this axiom follows commutativity of addition, which in turn proves uniqueness of zero.

If the things added are undefined, then the formula doesn't define anything specific.

Anixx
I do not see what definition you need. The axioms specify the equivalence classes of divergent integrals and operations on them.

I do not see what definition you need. The axioms specify the equivalence classes of divergent integrals and operations on them.

You haven't offered any mathematical definition of "divergent integral" yet.

The first step in developing an Algebra of Divergent Integrals should be to define precisely what a divergent integral is and to define the notation used to represent divergent integrals.

It is not correct to assume the definition of a non-divergent (definite) integral implies a specific definition for a divergent integral. The set ##S## of non-divergent integrals is, by definition, the set of numbers because each non-divergent integral is, by definition, equal to a number. To say that we have a thing ##A## that is not a member of ##S## tells us nothing about the properties of ##A##. For example ##A## could be a giraffe or the sentence "How are you, Mr. Wilson?".

I agree that you can define an algebra by simply postulating the existence of a set ##\mathbb{A}## who elements have certain operations with certain properties. But this approach does not define the connection that you are apparently making between ##\mathbb{A}## and pre-existing mathematical objects like functions and numbers. You apparently want to denote elements of ##A## with the pattern ##\int_a^b f(x) dx## where ##f(x)## is a symbol representing a real valued function and ##a.b## are symbols representing numbers or the symbols ##+\infty, -\infty##. So are you defining ##\mathbb{A}## to be the set consisting of all such patterns of symbols?

Anixx
What real number is? Usually it is defined as behavior of a sequence at infinity. Then defined which two sequences are equal and arithmetic operations. Here we have the same thing except instead of sequesnces we use smooth functions.

Anixx
But this approach does not define the connection that you are apparently making between and pre-existing mathematical objects
Convergent integrals and divergent integrals summable using a stable method (Cesaro, Abel, Borel) are considered equal to the values to which they are summable. This is in the axioms.

Convergent integrals and divergent integrals summable using a stable method (Cesaro, Abel, Borel) are considered equal to the values to which they are summable. This is in the axioms.

If your definition of divergent integral equates it to a real number, then how would an "algebra of divergent integrals" contain the real numbers as a proper subset? Why would an "algebra of divergent integrals" be an extension of the real numbers instead of a subalgebra of the real numbers?

By "the axioms", are you referring to the contents of the first post in the link you gave https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 ?

Anixx
Because real numbers are usually defined as convergent sequences of rational numbers, and divergent integrals or series are sequences or smooth functions that diverge at infinity.

and divergent integrals or series are sequences or smooth functions that diverge at infinity.

Convergent integrals and divergent integrals summable using a stable method (Cesaro, Abel, Borel) are considered equal to the values to which they are summable.

Sorry, but if you can't give a precise definition for a divergent integral, I can't suggest any way of defining an Algebra of Divergent Integrals. Perhaps some other forum member cares to offer a definition.

Anixx
Divergent integral is such integral whose partial Riemannian sum goes to infinity, of course.

Mentor
Divergent integral is such integral whose partial Riemannian sum goes to infinity, of course.
Not necessarily. This would also be a divergent integral, wouldn't it?
##\int_0^\infty \sin(x) dx##

In any case, I don't see the point in an algebra of integrals that diverge.

Anixx
That integral diverges, but it is Cesaro-summable, so it is equal to a real number (1).

Staff Emeritus
Gold Member
Not necessarily. This would also be a divergent integral, wouldn't it?
##\int_0^\infty \sin(x) dx##

In any case, I don't see the point in an algebra of integrals that diverge.

There's a huge field of physics that involves taking divergent integrals and getting numbers

https://en.m.wikipedia.org/wiki/Renormalization

Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics.

there are also other areas where performing algebra on divergent values can be useful. As one example, you can count the number of integer points in a polytope by writing down a polynomial and evaluating it at the point 1. It turns out the easiest way to do this (in some sense) is to write down infinite sums that don't converge at 1, but can be realized as simple functions like 1/(1-x), and then you can cancel out the singularities carefully and end up getting the right number.

There's a huge field of physics that involves taking divergent integrals and getting numbers

Whatever the practical uses of divergent integrals, the mathematical task of defining an algebra of such integrals (to currrent standards of rigor used in abstract algebra) requires formulating a precise definition of "divergent integral".

To a student of calculus, it is completely obvious what a divergent integral is. However, to say what it is with sufficient precision to use in doing abstract algebra is not simple. The definition of "definite integral" in calculus is precise enough to define a mathematical object only when the definite integral exists .

To make an analogy, suppose a person wishes to develop an algebra of arithmetic expressions that includes expressions that do not define numbers ( e.g "##\frac{2}{1-1} + \frac{0}{4-(3+1)}##"). The elements of such an algebra would be defined by saying ##X \in \mathbb{A}## if and only if ##X## is a pattern of symbols that obey a certain syntax.

Applying this to the case of divergent integrals, we could define an integration-template as a triple ##(a,b,f)## where each of ##a,b## is a number or one of the symbols ##{-\infty, \infty}## and ##f## is a function.

A divergent integral ##(a,b,f)## could be defined as integration-template such that (using the ordinary notation of Riemann integration) ##\int_a^b f(x) dx## does not exist.

I doubt that a useful algebra for the set of all integration-templates can be created. I think getting something useful requires assuming additional properties that restrict the integration-templates under consideration to a subset that has nice properties.

The original poster is assuming the intuitive idea of "divergent integral" is sufficient to define an element of an abstract algebra. His viewpoint seems to be that an "integral" is an attempted computation - i.e. it is not a mathematical object representing a number, but rather a mathematical object representing a process (i.e. an algorithm). His classification of "integrals" seems to be based on ways in which this process can produce a number, or produce a number in some alternative way to Riemann integration, or fail to produce a number by any of the allowed methods.

It may be possible define an abstract algebra on a set of algorithms. However, it requires going far beyond the calculus-level concept of integrals.