The Consolidated Math Thread

Every time someone has a math problem they post a new thread for it, it clutters up this subsection of the forum, and real enlightenment rarely takes place since only one or two people ever give their problem-solving insights. This thread is intended to change that by providing a centralized discussion of any aspect of mathematics that anyone is having trouble with, is interested in, or just wants to bone up on after being out of practice for a spell.

To start things off, I'll post one of the first complex analysis problems I ever did; this is a relatively simple problem, and in fact it is possible for anyone who knows integral calculus to solve it. I realize that most of your problems won't be this advanced, but I want to see who's up to it:

Prove that ∫ (e^imθ)*(e^-inθ) dθ

evaluated from 0 to 2π is equal to 2π when m = n, and 0 when m =/= n, where m and n are integers. Hint: e^iθ = sin θ + i cos θ. You can thank Leonhard Euler for that one.

I'm not really expecting anybody to solve that given the general age group here, but those of you who know your calculus and have chutzpah may want to give it a go, as this type of problem will allow you to cut your teeth on more abstract, higher-level math.

I know there are other math nerds out there, so come on in and we can collectively sharpen our MAD SKILLZ! (Xenogears reference ftw)

Edited in an attempt to make the problem slightly more readable; the forum doesn't allow superscripts, apparently.

Oh for ****s sake. Just make me more depressed at all my former knowledge of calculus, lost in the void between the crevices of my cerebellum .

ok I'm over it

Actually could you put spaces between some of the characters in that initial equation? It's a bit messy to look at and hard to read.

I feel kinda dumb asking this but does the variable "i" still stand for imaginary number? I don't recall doing calculus with those except maybe on an imaginary number line....



PS: I feel lonely on this forum board.

Yes, i denotes the imaginary unit. But beyond Euler's identity, you don't need any further knowledge of complex analysis to complete the problem.

So if I have any limit questions, I'll get help in this thread?

Originally Posted by Soren707

So if I have any limit questions, I'll get help in this thread?

not from me, maybe project

I haven't dealt with limits in 3 years.

And also haven't even begun this problem yet.

Care to entice me w/ a hint project?

Originally Posted by Soren707

So if I have any limit questions, I'll get help in this thread?

Sure. What do you need to know?

@morval: Think about the possible forms the integrand could take on depending on the values of m and n. If it helps, try writing the integral out on notebook paper; in case you aren't familiar with internet substitutes for mathematical notation, ^ denotes that the expression following it is an exponent.

Ok, here's the question.

Determine the limit graphically. Confirm algebraically:

lim as x approaches 0 of (5x^3 + 8x^2)/(3x^4 - 16x^2).

Would I just factor out 2 x's on the top and bottom and cancel them? I'm not sure how to do it algebraically or if it can be done. I know it is in it's indeterminate form (0 over 0 by substitution) but although you can't use substitution, how would you get it to not be undefined?

As a warning, hehe, I might have other limit questions. Thanks for the help!

For most situations in which you're asked to find the limit of an indeterminate form, you can use L'Hopital's rule. This allows you to take the derivative of both the numerator and the denominator of a rational function if the limit of this function is an indeterminate form; in the case you gave, after two differentiations the question becomes the limit as x approaches 0 of (30x + 16)/(36x^2 - 32). At this point it is no longer an indeterminate form, so you can substitute and obtain the limit of -1/2.

Be aware that if you run into a function which follows a nonterminating pattern of differentiation, L'Hopital's rule will fail, because the functions will never be differentiable into a form that is not indeterminate. In those cases you'll either have to substitute or perform some clever algebra.

Originally Posted by Project 2501

For most situations in which you're asked to find the limit of an indeterminate form, you can use L'Hopital's rule. This allows you to take the derivative of both the numerator and the denominator of a rational function if the limit of this function is an indeterminate form; in the case you gave, after two differentiations the question becomes the limit as x approaches 0 of (30x + 16)/(36x^2 - 32). At this point it is no longer an indeterminate form, so you can substitute and obtain the limit of -1/2.

Be aware that if you run into a function which follows a nonterminating pattern of differentiation, L'Hopital's rule will fail, because the functions will never be differentiable into a form that is not indeterminate. In those cases you'll either have to substitute or perform some clever algebra.

*Gasp* So all I had to do was use the power derivative on the top and bottom?! Neat! But are you sure I don't have to do the derivative of a fraction where I'd have to take the bottom multiply by the derivative of the top then vice versa?

No, the quotient rule is not required for this unless you have one rational function divided by another, and you can simplify it with algebra much of the time in that case.