Riemann Definition and 586 Threads

Homework Statement What does it mean by this: The cauchy riemann equations are never satisfied when x and y are different from zero and when x=y=0 . Looking at the example of f(z)= l z l = \sqrt{x^{2}+y^{2}} Homework Equations The Attempt at a Solution

we know that \Gamma (s)= \int_{0}^{\infty}dxe^{-x}x^{s-1} however every factor of the Riemann Zeta can be obtained also from a Mellin transform \int_{0}^{\infty}dxf(x)x^{s-1} =(1-p^{-s})^{-1} where f(x) is the distribution \sum_{n=0}^{\infty}x \delta (x-p^{-n}) is there any...

Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...

Can some please draw a comparison between Riemann Integration and normal definite integration in terms of accuracy.

The Riemannian curvature tensor has the following symmetries: (a) Rijkl=-Rjikl (b) Rijkl=-Rijlk (c) Rijkl=Rklij (d) Rijkl+Rjkil+Rkijl=0 This is surely trivial, but I do not see how to prove that Rijkl=-Rjilk. :( Thanks.

How would the world benefit from the Riemann hypothesis being solved? Mathematicians have been trying to solve this for over 100 years, but have been unable to due to it's mind-boggling complexity and difficulty. What would the world benefit if this theorem was to be solved?

Would the field of the number theory collapse or flourish if the Riemann Hypothesis is proved as true?

"not riemann integrable" => "not lebesgue integrable"?? Hi, In general if a function is not Riemann integrable does this mean the function is also not Lebesgue integrable? Why or why not? I know that if the the function is Riemann integrable then its Lebesgue integrable, but I can't find...

When we talk about "Hilbert space" in (undergraduate) QM, we are typically talking about the space of square-integrable functions so that we can make sense out of \int_{-\infty}^{\infty} |\psi(\vec r,t)|^2 d^3x. But are we talking about Riemann-integrable functions or Lebesgue-integrable...

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...

Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...

This could be the way to proof. remember, this is not a proof. today I found a clue to solution to Riemann hypothesis: Let it be Riemann zeta function :ζ(s) The proof that all the non trivial zeroes lie on the critical strip when s = 1/2 + it let us suppose there are other zeroes...

After reading about the Riemann Zeta Function on Wolfram Alpha (http://mathworld.wolfram.com/RiemannZetaFunction.html), it's still unclear to me how the Euler product formula is essentially equal to the limit of a p-series. Someone please enlighten me

Homework Statement Show that the function f: [0,1] -> R defined by: f(x) = 1, if x=1/k for some k f(x) = 0, else is Riemann integrable on [0,1] Homework Equations The Attempt at a Solution I attempted the problem using Cauchy's criterion but found that this function is...

Homework Statement Let A={1/n, n =natural number} f: [0,1] -> Reals f(x) = {1, x in E, 0 otherwise Prove f is riemann integrable on [0,1] Homework Equations The Attempt at a Solution Not quite sure, but I think supf = 1 and inf f= 0 no matter what partition you take, then...

1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...

Homework Statement Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain. Homework Equations Cauchy riemann equations Hint: Prove If f and f* are both analytic on D, then f is constant. The Attempt at a Solution I think f need be...

I was wondering if one of the consequences of the Elliott-Halberstam conjecture would imply the Riemann Hypothesis (RH) or the Generalized Riemann Hypothesis (GRH)? Or at least if there is a connection between the Elliott-Halberstam conjecture and RH or GRH? I ask because the...

Homework Statement How to use veilbein to calculate Riemann tensor, Ricci tensor and Ricci scalar? (please give me the details) de^a+\omega_{~b}^a\wedge e^b=0, R_{~b}^a=d\omega_{~b}^a+\omega_{~c}^a\wedge\omega_{~b}^c. The metric is...

Hello everyone. I was hoping someone could clarify this "heuristic" argument I found online. First, what is the analytic function they speak of and is its derivative difficult to compute? Second, does this look like a legit argument? : If you take the derivative w.r.t s of both sides of sum...

Maybe someone will find something interesting in this paper. They have a reference to some 1995 work by Alain Connes. I didn't have time to look into this very much. Maybe it's amusing and maybe not: http://arxiv.org/pdf/1012.4665v1

Lets say we have a function of a complex variable z , f(z). I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition. Can someone give me an example where the CR equations hold but the function is not...

For those not familiar with the term Fermi estimate/problem/question see here: http://www.vendian.org/envelope/dir0/fermi_questions.html http://en.wikipedia.org/wiki/Fermi_problem My question: Between the time that Riemann posed his famous question (in 1859) and now, how many hours have...

Riemann Integrable <--> Continuous almost everywhere? I ran across a statement somewhere in the forums saying that a function is Riemann-integrable iff it is continuous almost everywhere, i.e. if its set of discontinuities has measure 0. Is that right? What about the case of a function...

Homework Statement Rn=\sum(i*e^(-2i/n))/n^2, i=1 Identify this Riemann sum corresponding to a certain definite integral. Homework Equations The Attempt at a Solution I got till 1/n^2 [1/e^(2/n)+2/e^(4/n)+3/e^(6/n)...n/e^2] and that's it. To my understanding I should be...

Homework Statement Suppose that f:[a, b] → ℝ is a function that is zero for all x ∈ [a, b] except for the values x_1,x_2,…,x_k. Find ∫[a b](f(x)dx) and prove your result. Homework Equations Definition of a Riemann integrable function...

hmm...

A metric consistent with interval: \mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right) get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor. But I don't find thus...

Part 1. Homework Statement The problem literally states... " The Integral. limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2 " According to the teacher, the answer is 1/2. I don't know why or how to get there. Part 2. The attempt at a solution...

Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year. The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...

Last summer I took Semi Riemann Geometry lesson. Almost all the definitions in Semi Riemann geometry are with the same Minkowski geometry. I don't understand what is the different between Minkowski Space an Semi Riemann Space.

Homework Statement The problem given is: Show that the function f(x) = 1/sqrt(x) is integrable on the compact interval [0,1]. Homework Equations We are only allowed to use theorems, definitions, and properties that have been covered in class or are in the book. The ones I...

Can someone tell me what a riemann invariant is in general terms please? i can't find it anywhere! thanks, lav

no. of field equations and components or Riemann tensor?? Someone was trying to explain to me about curvature in space. From what I got from what they were saying doesn't make sense to me. I'm not sure I understand what the number of components, N, of R\alpha,\beta,\gamma,\delta when compared...

Consider the separation of the Riemann Zeta function in two terms \begin{flalign*} \zeta(s) &= 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + 5^{-s} + 6^{-s} + ... = & \\ &=(1^{-s} + 3^{-s} + 5^{-s} + 7^{-s} + 9^{-s}+ ... ) + ( 2^{-s} + 4^{-s} + 6^{-s} + 8^{-s} + ...)&=& \\ &= (1 - 2^{-s}) \zeta(s) +...

Hi, I'm stuck on this problem here about composite function, help is appreciated: Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function I was able to solve...

Sean Carroll: Lecture Notes on GR (2:20): Presumable to be a coordinate system it would have to exist at more than one point! Does he mean to define a Riemann normal coordinate system as a chart such that g_{\mu\nu} takes its canonical form and the first derivatives \partial_\sigma...

[b]1. If abs(f) is Riemann integrable on [a,b], then f is Riemann integrable on [a,b]. True or false (show work) [b]2. A function f is Riem Int iff f is bounded on [a,b], and for every epsilon>0 there is a partition P of [a,b] s.t. U(f,P)-L(f,P)<epsilon [b]3. I believe that this...

(Dis)proof of Riemann hypothesis,Goldbach,Polignac,Legendre conjecture I'm just an amateur and not goot at english ^^;

Background: Math: An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics. A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature. Given the existence of a connection with Riemann...

Homework Statement Hello …….. I have a question about a statement mentioned in the book “Introduction to tensor calculus and continuum mechanics” . it is :- Where the space (Vn) is Riemann space . Is this statement really true ? Homework Equations The Attempt at a...

This is my first time posting & I am not familiar with how to get all the correct math symbols or how to use Latex, so I did the best I could. Homework Statement Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on...

HERE http://vixra.org/pdf/1007.0005v1.pdf is my proposed proof of an operator whose Eigenvalues would be the Imaginary part of the zeros for the Riemann Hypothesis the ideas are the following* for semiclassical WKB evaluation of energies the number of levels N(E) is related to the integral of...

Basically from what I understand the integral of a function, say ∫x^2dx from say 0 to 1, can be represented as the supremum/infimum of the function values within each of a countably infinite number of vanishingly small intervals in the domain created by a countably infinite number of partition...

http://en.wikipedia.org/wiki/Riemann-Lebesgue_lemma Have I made a mistake when it looks to me that the Wikipedia proof on Riemann-Lebesgue lemma looks like nonsense? How are you supposed to use dominated convergence theorem there?

Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...

If I have a function c(x,Δx) that gives the area between x and x + Δx of a function. The area under the function can be given by: Sum from j = 0 to n-1 of c(b/n j,c/b) As n tends to infinity and b is the upper limit of integration. How can I convert this from a sum into a integral? I'm not...

Hi, thanks for the attention and excuse for my bad english. I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor. We have two kind of riemann tensor: type (3,1) Rikml type (4,0) Rrkml There are also some symmetry...

Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx I am really...

Homework Statement If a<c<d<b and f is integrable on (a,b), show that f is integrable on (c,d) Homework Equations The Attempt at a Solution I know that f is integrable on (a,b) iff for all e>0 there exists step functions g and h such that g \leq f1(a,b) \leq h and I(g-h) <e (...