Radius of convergence without complex numbers

rsq_a

Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented.

In examining the function
[tex]
f(x) = \frac{1}{1 + x^2}
[/tex]

we can derive the series expansion
[tex]
\sum_{n=0}^\infty (-1)^n x^{2n}
[/tex]

We note that the ratio test (which does not involve complex numbers) indicates that the series necessarily diverges if
[tex]
| -x^2 | > 1
[/tex]

or [itex]x > 1[/itex].

However, returning to the function f(x), we see that the point x = 1 is not at all special. How is the specialness of x = 1 explained without the notion of a complex number?

Perhaps we are misleading the reader by our choice of a series expansion about x = 0. Had we done a series expansion about x = 1, then we would have found that there is something special about the points [itex]x = 1 \pm \sqrt{2}[/itex]. But the question remains, how can this be anticipated a priori, without complex numbers?

* Note that in the language of complex numbers, this is explained simply by the fact that the distance to the nearest singularity, x = i, is 1. But this involves the notion of the Argand plane.

*DonAntonio*

Quote by rsq_a

Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented.

In examining the function
[tex]
f(x) = \frac{1}{1 + x^2}
[/tex]

we can derive the series expansion
[tex]
\sum_{n=0}^\infty (-1)^n x^{2n}
[/tex]

We note that the ratio test (which does not involve complex numbers) indicates that the series necessarily diverges if
[tex]
| -x^2 | > 1
[/tex]

or [itex]x > 1[/itex].

[tex]{}[/tex]

Well, in fact if [itex]\,x<-1\,\,\,or\,\,\,x>1\,[/itex]

However, returning to the function f(x), we see that the point x = 1 is not at all special. How is the specialness of x = 1 explained without the notion of a complex number?

Perhaps we are misleading the reader by our choice of a series expansion about x = 0. Had we done a series expansion about x = 1, then we would have found that there is something special about the points [itex]x = 1 \pm \sqrt{2}[/itex]. But the question remains, how can this be anticipated a priori, without complex numbers?

* Note that in the language of complex numbers, this is explained simply by the fact that the distance to the nearest singularity, x = i, is 1. But this involves the notion of the Argand plane.

I don't think you can explain this in a satisfactory way without complex numbers, and I can't understand very

well why would anyone without a knowledge of complex numbers be interested in this stuff, unless it is

a first-second year mathematics student who's taking some basic real analysis and hasn't yet studied complex one. If this

is the case one can simply say "wait until you study complex analysis to fully understand why the convergence radius of this stuff

cannot be enlarged anymore."

DonAntonio

rsq_a

Quote by DonAntonio

[tex]{}[/tex]

Well, in fact if [itex]\,x<-1\,\,\,or\,\,\,x>1\,[/itex]

I don't think you can explain this in a satisfactory way without complex numbers, and I can't understand very

well why would anyone without a knowledge of complex numbers be interested in this stuff, unless it is

a first-second year mathematics student who's taking some basic real analysis and hasn't yet studied complex one. If this

is the case one can simply say "wait until you study complex analysis to fully understand why the convergence radius of this stuff

cannot be enlarged anymore."

Well, this is a mathematics forum, so presumably, we can pretend we're interested in answering the question for the question's sake...

In any case, I think the question is fairly clear. The reason I ask is because I'm designing a lecture course and part of this requires me to explain the necessity of complex numbers. However, that being said, I'm not sure that complex numbers are even necessary. After all, complex methods, in many cases, are simply algebraic and geometric simplifications. For example, it's easy to represent a 90 degree rotation by multiplication by [itex]i[/itex] rather than through less elegant vector and matrix operations. However, everything in the physical world involves real numbers, so I'm expecting that for any question that requires a real answer, there exists a method (perhaps only approximate) that is restricted to real numbers.

I don't think you can explain this in a satisfactory way without complex numbers

I don't like this answer because complex numbers are an imaginary construct. They're simply algebraic and geometric relations. So there must be a way to remain in real space and explain series convergence.

Here is another question: in the below figure, you have two functions that are infinitely differentiable. The one in blue has infinite radius of convergence for a series expansion about x = 0, while the one in red has a radius of convergence of only 1. Is there an easy way to see this without complex numbers?

I am sorry if I'm not explaining it well; I admit that it's still fuzzy in my head.

*DonAntonio*

Quote by rsq_a

Well, this is a mathematics forum, so presumably, we can pretend we're interested in answering the question for the question's sake

I honestly don't think so. This forum is, if I'm not wrong, designed to try to address questions about mathematics in a rather wide meaning of the word, but it has to be about mathematics.

...

In any case, I think the question is fairly clear. The reason I ask is because I'm designing a lecture course and part of this requires me to explain the necessity of complex numbers. However, that being said, I'm not sure that complex numbers are even necessary. After all, complex methods, in many cases, are simply algebraic and geometric simplifications. For example, it's easy to represent a 90 degree rotation by multiplication by [itex]i[/itex] rather than through less elegant vector and matrix operations. However, everything in the physical world involves real numbers, so I'm expecting that for any question that requires a real answer, there exists a method (perhaps only approximate) that is restricted to real numbers.

It's weird, and if I may add a little worrying, to read the above from someone who is about to design a lecture course.

The importance of complex numbers cannot possibly be overestimated, and it's a fact that more than half of physics would fall apart

without them (yes: electricity, relativity, gases, mechanics, optics...all these use pretty heavy complex machinery).

To state that "everything in the physical world involves real numbers" is a huge understatement, in the best of the cases,

and simply a confusing and misleading, or even completely inaccurate, one in the worse of the cases.

I don't like this answer because complex numbers are an imaginary construct.

As imaginary as the numbers [itex] -4, 1.3, 7, 32[/itex] . ALL numbers are a mental abstraction of something. In some

cases we can easily grasp a rather "simple" abstraction, as saying that "5 is the number of fingers in my hand", and in

others it may well be pretty hard, but they all are abstractions and none, imo, is more imaginary or real than other.

This is one example more of long used poor names for things and here we have the result: people get a false idea.

They're simply algebraic and geometric relations. So there must be a way to remain in real space and explain series convergence.

Yes, there's a way to explain convergence of real series, but I can't see any way whatsoever to explain why

in some cases the convergence radius can be widened and why not, and what's wrong with the function [itex]\frac{1}{1+x^2}[/itex] staying all

the time within the real numbers realm.

Here is another question: in the below figure, you have two functions that are infinitely differentiable. The one in blue has infinite radius of convergence for a series expansion about x = 0, while the one in red has a radius of convergence of only 1. Is there an easy way to see this without complex numbers?

I am sorry if I'm not explaining it well; I admit that it's still fuzzy in my head.

I can't see any function in colors, but I'm afraid the answer will be closely related to the previous one.

DonAntonio

Ps. Ok, already say the little graph attached to your previous message. I can't say anything about

that until I see an analytical expression of the functions, of course.

rsq_a

Quote by DonAntonio

It's weird, and if I may add a little worrying, to read the above from someone who is about to design a lecture course.

The importance of complex numbers cannot possibly be overestimated, and it's a fact that more than half of physics would fall apart

Um. Thanks. I work with complex numbers every day (conformal mapping, contour integration, complex Fourier transforms, boundary integral methods, steepest descent approximations, etc.) ... so yes, I'm aware of their applicability, particularly as it relates to problems in mechanics.

without them (yes: electricity, relativity, gases, mechanics, optics...all these use pretty heavy complex machinery).

It doesn't mean that complex variables are necessary.

To state that "everything in the physical world involves real numbers" is a huge understatement, in the best of the cases,

Well of course it does. Any computation you use in designing a bridge, an airplane, in explaining the weather, will use real numbers. Now you may have to go through the complex numbers, but eventually the result is real.

It's like the solution of a cubic equation (which resulted in the birth of complex numbers, to an extent). Any cubic polynomial has at least one root, but the algebraic representation of the cubic roots may require formally manipulating negative square roots...however, once all the algebra is done, the solution is real.

Another example is Schrodinger's equation. Yes, it may be a complex differential equation, but eventually the only result we care about is real (namely, the amplitude and wavenumber of the wave function). You could have just as well decoupled the real and imaginary components of such equations. This is particularly evidenced in any numerical computation, where the computer treats imaginary numbers effectively as real vectors.

(Of course, I should add, this is more of a "I can't think of any physical problem that involves something more than real numbers", but perhaps I'm shortsighted).

and it's a fact that more than half of physics would fall apart

Can you give me a concrete example of a physics problem that would fall apart if I removed the complex plane?

micromass

Quote by rsq_a

Well of course it does. Any computation you use in designing a bridge, an airplane, in explaining the weather, will use real numbers. Now you may have to go through the complex numbers, but eventually the result is real.

So just because the result is real implies that complex numbers are unnecessary?? That's a weird form of reasoning.
You could also say that all results are rational numbers. Indeed, every measurement we do will be of the form 0.1432, which is rational. We can even work with the approximation 3.1415 for pi, nobody is going to notice the difference.
So, are real numbers unnecessary then?? Should we just do all math with rational numbers???

rsq_a

Quote by micromass

So just because the result is real implies that complex numbers are unnecessary?? That's a weird form of reasoning.
You could also say that all results are rational numbers. Indeed, every measurement we do will be of the form 0.1432, which is rational. We can even work with the approximation 3.1415 for pi, nobody is going to notice the difference.
So, are real numbers unnecessary then?? Should we just do all math with rational numbers???

Yes and no.

Your example isn't valid, because we need our number system to be dense. It's important that pi is irrational, otherwise all our (real) results would be off. I'm not talking about numerical precision. If you could theoretically compute to arbitrary precision, then it's relevant that the area of a unit circle is pi and not 3.14. So your sensationalist example is a bit of a strawman.

My statement (which, if you examine books on the development and philosophy of mathematics) is simply that, because complex numbers are effectively notational convenience applied to 2d geometric transformations, they're unnecessary. Basically, my understanding was that there is a real analogue for every complex result, in the same way that a complex number is nothing more than a vector in R2 with special geometric properties.

Some years back, I remember a chapter of such a book devoted to this issue, but unfortunately, the title eludes me.

Perhaps we should start from a concrete example. Can someone answer this question? If "half of physics falls apart" then surely one can easily come up with an example in which it's indisputable that complex numbers are necessary.

and it's a fact that more than half of physics would fall apart

Can you give me a concrete example of a physics problem that would fall apart if I removed the complex plane?

micromass

Quote by rsq_a

For example, it's easy to represent a 90 degree rotation by multiplication by [itex]i[/itex] rather than through less elegant vector and matrix operations.

Sure, you can do all that. But complex numbers are isomorphic to the rotation matrices. So you just used the complex numbers anyway, you just didn't call them that and you made it annoying for yourself.

Quote by rsq_a

You could have just as well decoupled the real and imaginary components of such equations.

Again, this is isomorphic to the complex number system.

rsq_a

Quote by micromass

Sure, you can do all that. But complex numbers are isomorphic to the rotation matrices. So you just used the complex numbers anyway, you just didn't call them that and you made it annoying for yourself.

Again, this is isomorphic to the complex number system.

Perfect. We're in agreement.

So how do you explain the notion of radius of convergence as the distance from the point of expansion to the nearest 'singularity' without the notion of complex functions and the complex plane (e.g. for example, in terms of geometry in R^2)?

For example, the expansion of 1/(1+x^2) about x = 1 should have radius of convergence |x - 1| < sqrt(2), right? How can this be seen entirely in terms of real numbers?

micromass

Quote by rsq_a

Your example isn't valid, because we need our number system to be dense.

The rationals are dense.

It's important that pi is irrational, otherwise all our (real) results would be off. I'm not talking about numerical precision. If you could theoretically compute to arbitrary precision, then it's relevant that the area of a unit circle is pi and not 3.14. So your sensationalist example is a bit of a strawman.

No, it's not a strawman. You said something like "the complex numbers are unnecessary because all results are real". Well, I say: "the real numbers are not needed because all results are rational".
The things you describe are tools to get to the eventual result.

Second, I can describe all results that use real numbers with rational numbers. And I really mean, every single result. It will be long and tedious though.

Basically, my understanding was that there is a real analogue for every complex result, in the same way that a complex number is nothing more than a vector in R2 with special geometric properties.

In the same way, there is a rational analogue for every real result.

The reals are just another human construct. It's not that the universe actually prefers real numbers. It's just something that humans invented. The complex numbers are exactly the same thing.

rsq_a

Quote by micromass

The rationals are dense.

No, it's not a strawman. You said something like "the complex numbers are unnecessary because all results are real". Well, I say: "the real numbers are not needed because all results are rational".
The things you describe are tools to get to the eventual result.

Second, I can describe all results that use real numbers with rational numbers. And I really mean, every single result. It will be long and tedious though.

Oops. I think I understand. So for something like the circumference of a circle, pi would be expressed as the limit of some rational sequence, correct? I'm happy with that. You're right and it was a good point.

The reals are just another human construct. It's not that the universe actually prefers real numbers. It's just something that humans invented. The complex numbers are exactly the same thing.

Okay, so I'd like to understand the notion of radius of convergence sticking with the real numbers. Nothing wrong with that question, right?

chiro

For rsq_a:

Why don't you just look at convergence in terms of the norm of R^2? This is typically how you deal with convergence criteria in any multi-dimensional space whether it's 2,3,4 and even infinite-dimensional spaces of orthogonal components (i.e. Cartesian-like geometry).

Once you get a constraint on the norm with regards to convergence, then you can look at the geometric interpretation of said norm.

rsq_a

Quote by chiro

For rsq_a:

Why don't you just look at convergence in terms of the norm of R^2? This is typically how you deal with convergence criteria in any multi-dimensional space whether it's 2,3,4 and even infinite-dimensional spaces of orthogonal components (i.e. Cartesian-like geometry).

Once you get a constraint on the norm with regards to convergence, then you can look at the geometric interpretation of said norm.

Yes, thank you for helping to get it back on topic.

I did think of it this way: consider the series expansions around x = 0 of
[tex]
f(x) = \frac{1}{1+x^2} \quad \text{and} \quad
g(x) =\frac{1}{1-x^2}
[/tex]

Although these two functions have differing values, their rate of convergence is identical (via the ratio test). It seems that what we need is to seek a 2d analogue of this, i.e. we need a function in [itex]R^2[/itex] that when expanded about (x,y) = (0,0) has a similar rate of convergence.

Maybe at this point, it requires the 'leap' that I can use
[tex]
h(x,y) = \frac{1}{1 - (x,y)^2}
[/tex]

But uh-oh, what does it mean to square a vector (x,y)? We define this analogously to complex arithmetic (I guess this requires a leap). Once I've done this, then everything follows from the usual series expansions.

I think this would work, and as you said, the key is to impose a restriction of an identical rate of convergence (but different series values). This allows us to develop the geometry which ultimately connects (1,0) and (-1,0) with the unit circle.

chiro

Well the unit circle is just the length of any vector in R^2 = 1.

Personally I think the best way to do this is to get a series expansion for the function in question and then just use the convergence results for when the addition of scalars converges (the scalars will relate to the norms).

Muphrid

What does it mean to square a vector? It means for a vector [itex]s[/itex], the square is [itex]s^2 = s \cdot s[/itex]. Easy enough.

To be honest, while complex analysis is a rich and storied subject, I think from a modern perspective it's a mistake to emphasize it as a separate discipline from vector analysis of the 2d Euclidean plane. All the usual results of complex analysis can be understood in terms of vector space theorems--Cauchy-Riemann condition, Cauchy integral theorem, all of it.

Case in point: if you have a vector field [itex]F(r)[/itex], then the Cauchy-Riemann condition is equivalent to saying [itex]\nabla \cdot F = \nabla \times F = 0[/itex]. This also makes the generalization of analytic functions to higher dimensional spaces obvious.

pwsnafu

Quote by Muphrid

Case in point: if you have a vector field [itex]F(r)[/itex], then the Cauchy-Riemann condition is equivalent to saying [itex]\nabla \cdot F = \nabla \times F = 0[/itex]. This also makes the generalization of analytic functions to higher dimensional spaces obvious.

Huh? You have a cross product. How is it obvious in higher dimensions?

*DonAntonio*

[QUOTE=rsq_a;4013990]Um. Thanks. I work with complex numbers every day (conformal mapping, contour integration, complex Fourier transforms, boundary integral methods, steepest descent approximations, etc.) ... so yes, I'm aware of their applicability, particularly as it relates to problems in mechanics.

It doesn't mean that complex variables are necessary. [QUOTE]

[tex]{}[/tex]

I'm not sure about this, but unless you can find an alternative I think that yes, it

actually does...

Well of course it does. Any computation you use in designing a bridge, an airplane, in explaining the weather, will use real numbers.

[tex]{}[/tex]

I think this is false but since I am not into engineering I can't tell for sure. What I can tell for sure is

that many calculations done in designing stem from complex numbers.

Now you may have to go through the complex numbers, but eventually the result is real.

[tex]{}[/tex]

This is blatantly false. Perhaps you meant to express that "eventually" you will use only the real part

of the solution of something in complex analysis, which I know for sure is also not the case in

many instances, but even then you had to resource to complex analysis to arrive to the solution

It's like the solution of a cubic equation (which resulted in the birth of complex numbers, to an extent). Any cubic polynomial has at least one root, but the algebraic representation of the cubic roots may require formally manipulating negative square roots...however, once all the algebra is done, the solution is real.

[tex]{}[/tex]

Again, this is false: the real cubic [itex]\,x^3-x^2+x-1[/itex] has one real roots but two complex non-real ones.

Even more striking: real equations of the form [itex]\,x^n-2\,[/itex] may have one single real roots and all the other n-1 ones

complex non-real, and if you change the minus for a plus and n is even then it may even be no real roots at all.

Another example is Schrodinger's equation. Yes, it may be a complex differential equation, but eventually the only result we care about is real (namely, the amplitude and wavenumber of the wave function).

[tex]{}[/tex] HUGE correction: Eventually you only care about the real solutions, and it strikes me as

you don't know a lot of deeper implications of some stuff, which is fine if you're interested onloy in some rather limited

applications, but it doesn't mean everybody does the same...not even close!

DonAntonio

You could have just as well decoupled the real and imaginary components of such equations. This is particularly evidenced in any numerical computation, where the computer treats imaginary numbers effectively as real vectors.

(Of course, I should add, this is more of a "I can't think of any physical problem that involves something more than real numbers", but perhaps I'm shortsighted).

Can you give me a concrete example of a physics problem that would fall apart if I removed the complex plane?

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chiro	Well the unit circle is just the length of any vector in R^2 = 1. Personally I think the best way to do this is to get a series expansion for the function in question and then just use the convergence results for when the addition of scalars converges (the scalars will relate to the norms).