Reddit mentions of Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory

>"The sum of two positive numbers is always greater than either number"

This "axiom" is a bit silly since you'd expect to be able to deduce what is essentially the triangle inequality from a more fundamental set of axioms. But hey listen, if you want to hold that as one of your own personal axioms and deduce some logical truths from that then more power to you. You can do math however the hell you want to do it.

(Quick side-note: I'd be careful in how you extend some of the "rules" for summation from finite to infinite sums because it doesn't always work! For example, while commutativity is a trivial property of finite sums, it turns out that commutativity does not hold (in general) when you consider infinite sums!)

>Fair enough. Which definition of limit would you like to use to justify the -1/12 limit?

The sum of the naturals does not approach -1/12 in any limit. Again, if you want to define "sum" to be the limit of the sequence of partial sums then thats totally fine. My point is that you can produce useful and powerful results by considering other more general meanings to the word "sum".

>That statement in quotation marks doesn't give you a basis to justify saying "sum of naturals is equal to -1/12". Of course, as humans and mathematicians we can attribute meaning wherever we find necessary.

Honestly I don't really understand what you mean here.

>Along the standard definitions of sums and convergence, yes people did.

You've heard people say that the sum of the naturals converges to -1/12? That's silly. You can prove that the sum of the natural is a divergent series very easily!

>As of right now, the whole idea of sum and equal to are undefined by you.

I gave you a whole list of "sums" to choose from! If you want me to pick just one then I'd perhaps pick the Zeta function regularization?

>And it's you whose trying to say that "nature thinks -1/12 is the sum of naturals". That's quite a bold statement.

I know, right! But at the end of the day its an experimentally validated statement. EDIT: I don't like how I worded this. Nature doesn't "think" anything, I think. It'd be more justified to say that QED analytically continues divergent sums to finite answers.

>I'm going to assume the scientists were using the standard definitions and that their theoretical value (sum of naturals) is infinity.

This just isn't true. Instead throwing up our hands and saying "oh, nature has given us a divergent series, time to look for a new theory" we figured out a way to make sense of this divergent answer. Whether you like it or not analytically continuing the zeta function and substituting in -1/12 for the sum of the naturals gave the right answer. Does that mean that the sum of the naturals converges to -1/12? No. Does that mean that the sum of the naturals approaches -1/12 in some limit? Of course not. Does it mean that the sum of the naturals equals -1/12? Reasonable people could say 'of course not, that goes against the entire field of mathematics!' OR 'well.. I just substituted in -1/12 for the sum of the naturals and I got the right answer. Perhaps there is some sense deep connection between these two quantities! Could a new definition of 'equality' yield more insight to this relationship?' Both are reasonable statements – I'm just saying that the latter is a more productive way of looking at things.

>Well I was under the impression that limits were calculus. Okay, what other meaningful areas of mathematics should we bring in here?

Mathematical physics. Wikipedia claims that renormalization is used in self-similar (fractal) geometry but I am not familiar with that field to know any examples.

>Keep in mind what you've been doing here. You've been saying there are other areas of mathematics and other definitions but not saying what specifically they are. I'd like you to start doing that please.

Hopefully I've 'name dropped' a sufficient amount of fields for you. If not then I would check out this book on a bit more information on the topic. Alternatively, if you're a fan of free youtube lectures, Carl Bender has an amazing lecture series on mathematical physics that sheds some light on the topic of assigning meaningful values to divergent series.

At the end of the day I'd like to reiterate my initial point: reasonable people can disagree about these topics and thats ok. My entire point here is that one should never be to extreme with their views on math (or really anything). If we applied your exact outlook on the sum of divergent series on the introduction to complex variables then we'd be doomed to have fields such as complex analysis or quantum mechanics forever out of reach. Being accepting of new outlandish ideas could lead to profound results!

This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.

This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.

This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.

This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.

Hope this helped, and good luck with your postgrad!