Hopefully this is the correct stack exchange for this, if not feel free to tell me where to move it where it would be more appropriate.
So I am a teaching assistant and masters student in the math dept. at a large state university. I was holding a discussion section this morning for the...
I imagine myself flipping a coin repeatedly and recording the outcomes. With only the WLLN being true, I expect to periodically encounter long strings of mostly heads or mostly tails, causing the running average to fall outside some epsilon's distance from the mean. These strings would occur...
Homework Statement
Show that there is no non-abelian group G such that Z(G)=\mathbb{Z}_2, which satisfies the short exact \mathbb{Z}_2\rightarrow G\rightarrow\mathbb{Z}_2^3.
The Attempt at a Solution
I have knowledge of group theory up through proofs of the Sylow theorems. I know the center...
Proving a Set is Closed (Topology)
Homework Statement
Let Y be an ordered set in the order topology with f,g:X\rightarrow Y continuous. Show that the set A = \{x:f(x)\leq g(x)\} is closed in X.
Homework Equations
The Attempt at a Solution
I cannot for the life of me figure...
Comparing Partions of a Natural Number
Homework Statement
Let r(n) denote the number of ordered triples of natural numbers (a,b,c) such that a + 2b + 3c = n, for n\geq 0. Prove that this is equal to the number of ways of writing n = x + y + z with 0\leq x \leq y \leq z for x,y,z natural...
Given the domain ℂ\[-1,1] and the function, f(z)=\frac{z}{(z-1)(z+1)}, defined on this domain, the Residue Theorem shows that for \alpha a positive parametrization of the circle of radius two centered at the origin, that:
\int_{\alpha}f(z)=\int_{\alpha}\frac{z}{(z-1)(z+1)} = 2\pi i
Can I...
I'm trying to prove that log|z| is not the real part of an analytic function defined on an annulus centered at zero. Due to the Cauchy-Riemann Equations, I've been under the impression that given a harmonic function, such as log|z|, its role as the real part of an analytic function is unique...
Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a...
Homework Statement
Evaluate the integral
\int_0^{2\pi}log|e^{i\theta}-1|d\theta
Homework Equations
The Attempt at a Solution
So I'm essentially integrating log|z| around a circle of radius 1 centered at -1. Evaluating at the endpoints gives a singularity, but I feel like that...
I vaguely recollect that the following statement is true:
Let f be analytic on a connected set D, then if f is constant on some nonempty open subset of D then it is constant on all of D.
Can anyone confirm that this is true and is it a specific theorem? Thanks.
Homework Statement
Show that:
\frac{1}{2\pi}\int_0^{2\pi}log|re^{i\theta} - z_0|d\theta = \left\{\begin{matrix}
log|z_0| & if & |z_0| < r \\
log|r| & if & |z_0| > r
\end{matrix}\right.
Homework Equations
The function log|z| is harmonic in the slit plane since it is the real part of...
Homework Statement
Prove |\frac{e^{2i\theta} -2e^{i\theta} - 1}{e^{2i\theta} + 2e^{i\theta} -1}| = 1
Homework Equations
The Attempt at a Solution
I feel like this should be fairly simple, anyone have any hints? Also this is just one step in an attempt to solve a much larger problem, so...
I need a quick reminder that this is (hopefully) true:
Let \sum a_n be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts?
\sum a_n = \sum x_n + i\sum y_n
Homework Statement
Let f(z) be a complex function analytic everywhere except at a where it has a singularity. Prove that the function f(z) - \frac{b_{-1}}{z-a} has a primitive in a punctured neighborhood of a. Where b_{-1} is the coeffecient of the n=-1 term in the Laurent expansion of f(z)...
So I have this statement that I'm supposed to prove and I cannot for the life of me figure out what parts I'm allowed to assume and what part I am expected to prove, here it is:
The residue of an analytic function f at a singularity a ∈ ℂ is the uniquely determined complex number c, such that...
Edit: Never mind I found my error, moderator can lock this.
Homework Statement
Evaluate the integral \int_0^{\pi} \frac{dt}{(a+cost)^2} for a > 1.
Homework Equations
\int_0^{\pi}\frac{dt}{(a+cost)^2} = \pi i\sum_{a\epsilon \mathbb{E}}Res(f;\alpha)
Where \mathbb{E} is the open unit...
I've got this complicated expression that I'm trying to simplify and this is one piece which I feel might have a really simple form: ((i+1)^{n+1} - (i-1)^{n+1)) for n ≥ 0. Thanks.
Say we have two power series \sum_{n=0}^{\infty}a_n z^n and \sum_{n=0}^{\infty}b_n z^n which both converge in the open unit disk. Is there anything we can say about the radius of convergence of the power series formed by their difference? i.e. \sum_{n=0}^{\infty}(a_n-b_n) z^n
What about if we...
Homework Statement
Find the impulse and frequency responses of the following systems:
1. y(n) = \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})x(n-k)
2. y(n)=ay(n-1)+(1-a)x(n), where 0<a<1
Homework Equations
The Attempt at a Solution
Ok so for 1. I look at h(n) which is...
Homework Statement
Let D ⊂ C be open, connected, and bounded. Suppose the boundary of D consists of a finite number of piecewise differentiable simple closed curves: α0,...,αN, with α1,...,αN contained in the interior of α0. Suppose α0 is oriented in the positive direction and α1, . . . , αN...
Ok I can do the integral and see that it is equal to 2∏i, but thinking about it in terms of 'adding up' all the points along the curve I feel like every every point gets canceled out by its antipode, e.g. 1/i and -1/i.
I'm curious if 1/x ~ 0. Technically by the definition I know it's not since lim x→∞ (1/x)/0 = ∞. But I feel like it does satisfy what the 'on the order of twiddles' is trying to measure. Thus I was wondering if maybe we specially define this to be true in the same way we might define 0! = 1.
This seems to be the new 'it' job in the tech sector and I'm considering getting into this line of work, but because it seems relatively new I'm having trouble finding out what it's like to work as a data scientist. The two things I'm somewhat concerned about is if the work is interesting or...
I stumbled across this link http://www.resumeserviceplus.com/advices.php?topic=5-Highest-Paying-Engineering-Specializations which gives data from the US Bureau of Labor Statistics on engineering salaries.
The second table breaks things down by education level. Can someone explain to me why...
Homework Statement
Let f be entire. Then if lim_{z\rightarrow \infty}|f(z)|=\infty then f must be a non-constant polynomial.
Homework Equations
The Attempt at a Solution
So we know f is entire. Thus I suppose it makes sense to go ahead and expand it as a power series centered at zero...
So given \int_c^d \int_a^b f(x,y)dxdy, we can exchange the order of the integrals provided that \int_c^d \int_a^b |f(x,y)|dxdy < \infty. Does this less-than-infinity property have to hold for both orders of iteration i.e. for dxdy and dydx? Or can it be proven that if it's finite for one order...
Homework Statement
Let f be a suitably regular function on ℝ. (whatever that means).
What function do we obtain when we take the Fourier transform of the Fourier transform of f?
Homework Equations
F(s) = \int_{x=-\infty}^{\infty}f(x)e^{-2\pi isx}dx
The Attempt at a Solution...
Say we have a complex function f, analytic on some punctured open disk D\{a} where it has a pole at a. Is there some theorem which says something like: f must map D\{a} to a horizontal strip in ℂ of at least width 2π, or something like that?
Homework Statement
If a 3-digit number (000 to 999) is chosen at random, find that probability that exactly 1 digit will be >5
The Attempt at a Solution
So basically I first look at the probability of at least 1 digit being greater than 5, taking into account multiple counting:
P(A ∪ B ∪ C)...