Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Notes on Taylor and Maclaurin Series

eli.thegreenplace.net

Post reply on HN
Print view View original

11–20 of 34 posts

Re: Notes on Taylor and Maclaurin Series

#11

Post by curt15 »

btilly wrote: post #9

In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series. Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down. I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like,…

Instead of thinking of real analysis as a zoo of weird exceptions, it's probably more accurate to think of complex analyic functions as the exceptions. For example, when viewed as a two dimensional mapping from the plane to itself, complex-analytic functions are conformal (angle-preserving) whereas most differentiable mappings from the plane to itself are not.

Re: Notes on Taylor and Maclaurin Series

#12

Post by abdullahkhalids »

zamadatix wrote: post #2

All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :). The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.

Weird. When I was doing boxing, having to run sprints every training session, doing dozens of pushups, situps and what not did not beat the interest I had in boxing :).

Re: Notes on Taylor and Maclaurin Series

#13

Post by [deleted] »

btilly wrote: post #9

In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series. Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down. I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like,…

[deleted]

Re: Notes on Taylor and Maclaurin Series

#14

Post by borroka »

abdullahkhalids wrote: post #12
zamadatix wrote: post #2

All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :). The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.

Weird. When I was doing boxing, having to run sprints every training session, doing dozens of pushups, situps and what not did not beat the interest I had in boxing :).

Not at all weird, because you were also boxing. But if your boxing sessions were only to be sprints, pushups, situps etc, and no boxing, you would surely lose interest in doing boxing at that place. Likewise, if your jiu jitsu training were only drills and no sparring, there would be no people doing jiu jitsu after a while.

Re: Notes on Taylor and Maclaurin Series

#15

Post by dhosek »

curt15 wrote: post #11
btilly wrote: post #9

In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series. Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down. I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like,…

Instead of thinking of real analysis as a zoo of weird exceptions, it's probably more accurate to think of complex analyic functions as the exceptions. For example, when viewed as a two dimensional mapping from the plane to itself, complex-analytic functions are conformal (angle-preserving) whereas most differentiable mappings from the plane to itself are not.

On the other hand, complex analysis explains things that aren’t obvious in real analysis, like why the radius of convergence for a continuous function on the real line might be 1 (turns out there are singularities off the real line on the complex plane).

Re: Notes on Taylor and Maclaurin Series

#16

Post by abdullahkhalids »

borroka wrote: post #14
abdullahkhalids wrote: post #12

Earlier quoted context omitted.

Weird. When I was doing boxing, having to run sprints every training session, doing dozens of pushups, situps and what not did not beat the interest I had in boxing :).

Not at all weird, because you were also boxing. But if your boxing sessions were only to be sprints, pushups, situps etc, and no boxing, you would surely lose interest in doing boxing at that place. Likewise, if your jiu jitsu training were only drills and no sparring, there would be no people doing jiu jitsu after a while.

My boxing training sessions were about 50-50 exercises and the padding/sparring session (in which there were big breaks).

In my undergrad, I was made to take 6 math courses, and about 18 physics courses which used that math [1]. Plus additional Engineering/CS electives. So over a 25-75 ratio of exercise to "real" stuff.

Much better ratio than boxing.

[1] If you decide that the math courses have nothing real/interesting in them. Which is not true.

Re: Notes on Taylor and Maclaurin Series

#17

Post by kaashif »

btilly wrote: post #9

In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series. Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down. I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like,…

Complex analysis is far too nice. There are so many theorems where my first thought was "there's no way that's true".

Picard's great theorem is totally insane. As is even its little brother, and even Liouville's theorem.

The proofs aren't even that long. They just feel totally false.

Re: Notes on Taylor and Maclaurin Series

#18

Post by antognini »

One of the things that has always seemed rather magical to me is the Taylor series of the exponential function for very large negative values. We know that a number like e^-100 is a number extremely close to zero. Yet when you write out the Taylor series you see an alternating some of increasingly large, seemingly arbitrary numbers (at least for ~100 terms):

e^-100 = 1 - 100 + 5000 - 166,666.6 + 4,166,666.6 - ...

If you were just given this sum and knew nothing about Taylor series or the exponential function you'd assume that its value was some extremely large number. Yet everything manages to cancel out just right so that the resulting sum is almost exactly zero, but not quite.

I can't help but wonder if there's some parallel to parts of quantum field theory. If you expand out the interactions between two particles you also end up with a series that is apparently divergent. Yet we know experimentally that the first few terms work quite well as an approximation. It feels a bit like you're looking at the series of e^-100 without knowing about the exponential function.

Re: Notes on Taylor and Maclaurin Series

#19

Post by mkarrmann »

antognini wrote: post #18

One of the things that has always seemed rather magical to me is the Taylor series of the exponential function for very large negative values. We know that a number like e^-100 is a number extremely close to zero. Yet when you write out the Taylor series you see an alternating some of increasingly large, seemingly arbitrary numbers (at least for ~100 terms): e^-100 = 1 - 100 + 5000 - 166,666.6 + 4,166,666.6 - ... If…

Why do you think a series which absolutely converges but whose first few terms are a poor approximation is related to a series which diverges but whose first few terms match the empirically expected value?

Not only is it a stretch to wonder about parallels between a basic Calc II concept and Quantum Field Theory, but is seems like the exponential function is the exact opposite of the example you provided.

Re: Notes on Taylor and Maclaurin Series

#20

Post by tasteslikenoise »

antognini wrote: post #18

One of the things that has always seemed rather magical to me is the Taylor series of the exponential function for very large negative values. We know that a number like e^-100 is a number extremely close to zero. Yet when you write out the Taylor series you see an alternating some of increasingly large, seemingly arbitrary numbers (at least for ~100 terms): e^-100 = 1 - 100 + 5000 - 166,666.6 + 4,166,666.6 - ... If…

The whole polynomial functions that give the truncations of the exponential Taylor series are also well-behaved in a way that's really surprising if you haven't thought about it before. For example, if you plot 1, 1 + x + x^2/2, 1 + x + x^2/2 + x^3/6 + x^4/24, and so on, you'll see that they actually don't oscillate at all as you might expect for negative inputs (for the reason you're saying), but rather they're very nice convex "bowl-shaped" positive functions, which hug zero more and more closely along the negative axis as you include more terms. One way to get some intuition about this is to notice that the derivatives of the next polynomial in the sequence are just the ones that came before (related to exp(x) being its own derivative). In particular, if all the even degree ones so far have been convex, then so will the next one be.

The idea you mention at the end is not specific to something mathematically advanced like quantum field theory. Useful truncations of divergent series show up in many more elementary situations in calculus and applied math; here is a nice class taught by the excellent Steven Strogatz where he talks about a simple example in the first lecture:

https://www.youtube.com/watch?v=KZsk8B_z8pI&list=PL5EH0ZJ7V0...

Post reply on HN