Homework Statement
Show that
\frac{\sin (az)}{\sin (\pi z)} = \frac{2}{\pi} \sum_{n=1}^{+\infty} (-1)^n \frac{n \sin (an)}{z^2 - n^2}
for all a such that - \pi < a < \pi
Homework Equations
None really, we have similar expansions for \pi cot (\pi z) and \pi / \sin (\pi z) , this...
I'm sorry, I didn't realize my question was ambiguous.
But if there are no significant differences between metric and topological spaces, then your understanding is correct. Thank you for your answers. I'll try to be more specific in the future.
Well, I'm out of my league here. I'm only in my first undergrad year and I haven't taken any topology yet (although I have Munkres' book and intend to put it to good use in the summer), so I'll have to study some before a proof or counterexample along those lines will make sense to me.
This...
Does that equation have any real solutions? Or even complex ones?
We know that for all x that |cos x| <= 1, and therefore |cos^n x| <= 1. So based on that we get
cos x + cos^2 x + cos^4 x <= 1 + 1 + 1 = 3 < 4
so we must conclude that the original equation has no solutions.
By the way, I first tried arguing that since f sends compact sets into compact sets, f also sends closed sets into closed ones. This doesn't hold because it says nothing about a closed set which isn't totally bounded and all bets are off on that one.
Unless are no closed sets which aren't...
Hmm... that's strange. I thought I managed to prove this earlier. Here's how:
Let (M,d) and (N,r) be metric spaces, and f:M -> N a one-to-one and onto function. Assume that for every subset K of M holds
K compact in M <=> f(K) compact in N
Let's show that f is continuous.
Take a...
Hi there.
I'm taking a course in analysis and I was thinking about the relation between compact sets and homeomorphism. We know that if f is an onto and one-to-one homeomorphism then it follows that for every subset K:
K is compact in M <=> f(K) is compact in N
Now, does this go the...
There's another fun variation on this theme where you line up all the numbers from one to nine in threes and are supposed to make them add up to six by adding only plus, minus, division, multiplication, root and power signs (whole powers and roots, no logs!). You can also use ( and ) (forgot...
I'm starting university to learn mathematics and I'm looking for a good graphical calculator, what are good value-for-money models that would be useful for some time to come?
Thanks,
Gunnar.
Hello Nick,
Here's something that might interest you, it's an article from Sky and telescope magazine about the possibility of habitable moons in orbit around gas giants. http://skyandtelescope.com/resources/seti/article_255_1.asp
Enjoy,
Gunnar.
The last example on my homework assignment this week is this: Solve the following differential equation.
y'' + 2y' + y = e^{-x}
I started by solving it like it was y'' + 2y' +y = 0 (Instert y = e^ax and so on) and got the following equation and solution:
e^{ax}(a^2 + 2a + 1) = 0 => a...
Hmm... I did it by hand (subsitution, t = g + 1/2 kA/m v^2) and got this:
y = -\frac{m}{kA}ln(1 + \frac{kAD}{m})
Which is the same as mathematica returned, but with an extra minus sign. Is it an error on my behalf?
Oh, I see. That's odd. Are you sure that in the text they didn't mean that the result from the arclength formula should be x^2/2 + lnx/4 before you put in the limits?
I don't see the problem, I don't get an extra 1/2 at the end. I get the same result as you do finding the function but I don't have any problem inputing back into the equation.
f(x) = x^2 - \frac{ln|x|}{4}
\frac{df(x)}{dx} = x - \frac{1}{4x}
\frac{df(x)}{dx}^2 = x^2 - \frac{1}{2} +...
If I recall correctly from last year's biology plants don't know which way's up and which way's down. But the plant growth hormone (Axin or something like that) gathers at the bottom of the seed (pulled down by gravity) and that causes the plant to "know" which way's up. The plants roots then...
I'm sorry, I'm from Iceland and math is tought in icelandic, so I don't know all the english names for functions.
I'll look over what NateG posted tomorrow (kind of late here at GMT), since I've run into trouble deriving it myself the least I can do is learn how to do it properly. Thanks guys.
Quick question.
sin(-x) = -sin(x)
this can be seen as this example:
sin(i^2x) = i^2sin(x)
Does this then apply?
sin(ix) = isin(x) ?
I'm trying to derive a formula for the length of a simple parabola. Unfortunetly in my calculations I end up with Arcsin(i2x) along the way and it...
One year.
We were debating a similar question in my physics class a couple of weeks ago. The difference was we were thinking that if you had a rigid pole in the pole/barn paradox, you'd be both outside the barn and inside at the same time. We argued about it for a very, very long time until...
Actually the objeect isn't falling faster than g. It's speed is increasing but g (the accelaration) stays the same. It's no problem to fall faster than g, for example if you throw yourself out of an airplane you will be accelarating at a constant 9.8 m/s^2, one g (If there was no air to provide...
It's the same in both cases, I beleive. In the first, the work done equals F*(s_2 - s_1). In the second, the man can never be sure if he's moving towards the chair or the chair towards him if it/he is moving at a constant velocity, so it can be treated exacly like the first example. Work then...