# Search results

1. ### Bizarre incident today in the class I was teaching

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...
2. ### How would the world be different if only the WLLN were true?

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...
3. ### Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping

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...
4. ### Proving a Set in the Order Topology is Closed

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...
5. ### Show That the # of (a,b,c) s.t. a+2b+3c = n is same as # of x + y + z = n s.t. x≤y≤z

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...
6. ### F has a primitive on D ⊂ ℂ ⇒ ∫f = 0 along any closed curve in D?

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...
7. ### Determining an Analytic Function from its Real Part

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...
8. ### Help Me Prove this Identity (or find a counterexample)

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...
9. ### Tough Integral

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...
10. ### Trying to remember what Theorem this is.

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.
11. ### The integral of a harmonic function

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...
12. ### Proving Modulus of Rational Expression is Equal to 1

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...
13. ### Splitting Infinite Series into Real and Imaginary Parts

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
14. ### Proving this meromorphic function has a primitive in punctured plane

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)...
15. ### Help me parse the logic of this statement

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...
16. ### Evaluate real integral using residue theorem, where did I go wrong?

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...
17. ### Help me simplify this

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.
18. ### Subtraction of Power Series

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...
19. ### Check Understanding Impulse & Frequency Response

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...
20. ### Winding Number iff statement

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...
21. ### Why is integral of 1/z over unit circle not zero?

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.
22. ### Is 1/x ~ 0?

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.
23. ### Closed Curves on the Riemann Sphere

Is the imaginary axis considered a closed curve on the Riemann Sphere?
24. ### Anyone work as a Data Scientist/Data Miner/in Machine Learning?

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...
25. ### Why does a PhD in CompE command so much more \$

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...
26. ### Unbounded Entire Function must be Polynomial

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...
27. ### Question about Fubini's Theorem

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...
28. ### The Fourier transform of the Fourier transform

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...
29. ### Looking for an analytic mapping theorem

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?
30. ### From 000 to 999, probability exactly 1 digit is >5

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)...