What is Singularities: Definition and 178 Discussions

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic p it is an open problem in dimensions at least 4.

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  1. C

    Singularities in the harmonic oscillator propagator

    Hi people! Today I was doing some QFT homework and in one of them they ask me to calculate the Harmonic Oscillator propagator, which, as you may know is: W(q_2,t_2 ; q_1,t_1) = \sqrt{\frac{m\omega}{2\pi i \hbar \sin \omega (t_2-t_1)}} \times \exp \left(\frac{im\omega}{2\hbar \sin \omega...
  2. J

    Big Bang : Singularity or Singularities?

    The Big Bang is often associated with the concept of a singularity. A singularity is defined as a point in space-time. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in an Euclidean space. This seems to me very misleading in as much...
  3. twistor

    A couple of questions about singularities

    Which is the difference between a cosmological and a gravitational singularity? Is there any mathematical tool that relates them? Do they REALLY exist or are they of a nature that simply escapes GR? How can the Big Bang and flatness of space coexist? Could the Big Bang be a local event? How does...
  4. M

    I still don't understand singularities

    I still don't understand singularities! I'm sorry for anyone bored of reading the threads I've made so far but either I simply do not understand it or I'm imagining the singularity totally wrong. If I'm chosing my words correctly, at the centre of a black hole is a dimensionless point of...
  5. L

    Logarithmic singularities are locally square integrable

    Homework Statement I will like to show that the function f:\mathbb{R}^2\rightarrow \mathbb{R} defined by f(x)=\ln\bigg(1+\dfrac{\mu}{|x-x_0|^2}\bigg),\quad\mu>0 is in L^2(\mathbb{R}^2). Homework Equations A function is in L^2(\mathbb{R}^2) if its norm its finite, i.e...
  6. S

    Identifying singularities of f and classifying them

    Hi guys, just wanting to know if I'm doing this right. f(z) = \frac{z}{(z^2 + 4) (z^2+1/4)} Singularities of f(z) are when (z^2 + 4), (z^2 + 1/4) = 0 In this case, the singularities are \pm2i , \pm\frac{i}{2} Lets call these singularities s and s is a simple pole if \lim_{z...
  7. N

    MHB Residues/Classifying singularities

    Hi, so first of all I am not entirely confident with the terminology when it comes to classifying singularities. Could someone give me an example of the different types, or explain what they mean? My confusion stems from the terms: essential singularity, isolated singularity, removable...
  8. TrickyDicky

    Are singularities part of the manifold?

    Mod note: Posts split off from https://www.physicsforums.com/showthread.php?p=4468795 Hi, WN, might the OP be referring to GR instead of SR, more specifically to the expanding FRW universe in which it is impossible to even consider the notion of exansion without agreeing about an "everywhere...
  9. T

    Isolated Singularities: Removable Singularities and Poles

    I'm working on a few questions for an assignment but am unsure whether my approach to this type of question is sufficient or valid. I will show my solutions to two problems that are not part of the assignment just to ensure my method is correct. "...locate each of the isolated singularities...
  10. R

    On the regular singularities of a second order differential equation

    Homework Statement The only singularities of the differential equation y''+p(x)y'+q(x)y=0 are regular singularities at x=1 of exponents \alpha and \alpha', and at x=-1 of exponents \beta and \beta', the point at infinity being an ordinary point. Prove that \beta=-\alpha and \beta'=-\alpha'...
  11. franciobr

    Numerical integration - Techniques to remove singularities

    Hello everyone! I am trying to understand why the following function does not provide problems to being computed numerically: ∫dx1/(sin(abs(x)^(1/2))) from x=-1 to x=2. Clearly there is a singularity for x=0 but why does taking the absolute value of x and then taking its square root...
  12. V

    Cauchy-Goursat's theorem and singularities

    Homework Statement Calculate the closed path integral of \frac{z+2 i}{z^3+4 z} over a square with vertices (-1-i), (1,i) and so forth. Homework Equations The closed line integral over an analytic function is 0 The Attempt at a Solution Alright, so first I factored some stuff...
  13. M

    Substitution of variables to remove singularities.

    Homework Statement I am given an integral for which I need to substitute variables to remove a singularity so that the integral can be computed in Matlab using the Composite Trapezoidal Method, and then compared to the integral computed in Maple to 16 digit precision. I am stuck on the variable...
  14. H

    Singularities of two variables rational functions

    Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The Attempt at a...
  15. P

    MHB Singularities, Residues, and Computation: Analyzing $f(z)$ without Prefix

    Consider $$f(z)=\frac{z+5}{e^\frac{1}{z}-3}$$ Find and classify its singularities and compute residues. I think singularities are: $0,\infty$ and zeroes of denominators. We have $e^{\frac{1}{z}}=3$ for $z=(log(3)+2k\pi i)^{-1}$. I think $0$ is essential, $\infty$ a simple pole and zeroes of...
  16. S

    What are the Singularities of f(z) = log(1+z^1/2)?

    Homework Statement Find all the singularities of f(z)=log(1+z^{\frac{1}{2}}) Homework Equations Well I need to expand this. Find if it has removable singularities, poles, essential singularities, or non-isolated singularities. The problem is the branches. I know z^{\frac{1}{2}} has...
  17. zonde

    Singularities and Rindler horizon

    I am trying to understand things around singularities and related to this I have a question. What kind of singularity is Rindler horizon? Wikipedia (Rindler coordinates) says that: "The Rindler coordinate chart has a coordinate singularity at x = 0," But if Rindler coordinates are not...
  18. E

    Kln theorem and initial-state singularities

    Hi! If I have a pair q\bar q g in a final state, I know that the gluon has a IR singularity. But KLN theorem rescues me: if I sum over all degenerate states the IR singularity cancels away. Otherwise, if the emission of the soft gluon is in an initial state, then the IR divergence cannot be...
  19. C

    Contour Integration with Singularities: How Does the Residue Theorem Apply?

    For the purposes of complex integration with the residue theorem, what happens if one or more of the poles are on the contour, rather than within it? Is the residue theorem still applicable?
  20. bcrowell

    Naked singularities, timelike singularities

    Is there a logical connection between the concept of a naked singularity and the concept of a timelike singularity? On a Penrose diagram, black hole and big bang singularities are always spacelike. Global hyperbolicity (Hawking and Ellis, p. 206) basically means two conditions: (1) no CTCs...
  21. B

    Determine the singularities and evaluate residues

    Homework Statement f(z) = \frac{z*exp(+i*z)}{z^2+a^2} Homework Equations Res(f,z_0) = lim_z->z_0 (1/(m-1)!) d^{m-1}/dz^{m-1} {(z-z_o)^m f(z)} The Attempt at a Solution I have no clue how to do this because I don't know how to determine the order of the pole for a function of...
  22. T

    MHB Singularities of Complex Functions

    Determine the location and nature of singularities in the finite z plane of the following functions: (a) f(z) = ( - 1) sin(z)/[z(z+1)(z+2)(z-3)] (b) g(z) = [1 + cos(z)]/ Using Cauchy's intergral formulae, referring to the above functions, Evaluate i) f(z) dz, with C : | z + j | = 4 ...
  23. R

    About applicability of singularities in Physics

    Hello, I am new here and this is my first post. Kindly let me know if my post is off topic. My question is about the applicability of singularities of a function in Physics. By singularity I mean one of the higher derivatives (>2) of a function jumping at a point. Is there any conceptual use...
  24. K

    Solving Complex Integration Involving Bessel, Singularities

    Well, here it is. I am at a loss as to how to approach this. I understand I can use the residue theorem for the poles at a and b, those are not the problem. I have heard that you can expand the function in a Laurent series and look at certain terms for the c term , but I don't fully understand...
  25. G

    Where are the singularities of f(z) = Log(2+tan(z)) located?

    I need to find the locations of the singularity of f(z) = Log(2+tan(z)). So far I have looked at the function in its alternate form Log(2+tan(z)) = ln(abs(2+tan(z))) + i*Arg(2+tan(z)) If I remember correctly the first part is simple and cannot equal zero. Now I think the second...
  26. D

    Residue calculus for essential singularities

    Homework Statement Im not sure if i understood correctly how to calculate the residue for functions with essential singularities like: f(z)=sin(1/z) h(z)=z*sin(1/z) j(z)=sin(1/z^2) k(z)=z*(1/z^2) Homework Equations So, according to what I've read, when we have a functions with an essential...
  27. A

    Analyzing Singularities at z=2 & -1/3

    Homework Statement I have been asked to state the precise nature of the singularities at z=2 and z=-1/3 in Homework Equations I know the laurent series is given by The Attempt at a Solution I think I need to expand the series out into a laurent series around z=2 and z=-1/3 but...
  28. G

    Splitting an integral range to handle singularities

    I am required to write a program that uses Simpson's rule to evaluate ∫t**-2/3(1-t)**-1/3 dt from limits t=0 to t=1. The questions gives a hint to split the integral into two parts and use a change of variable to handle the singularities. I really don't know where to begin. Is the choice of...
  29. S

    Confusion about quantum foam, quantum gravity and singularities

    Hi all, I've just finished reading a book about Black holes (Black holes & time warps, by Kip Thorne) and there's something in particular I'm confused about. One chapter talks about what can possibly be inside the singularity, specifically what would happen to an astronaut as he falls...
  30. fluidistic

    Singularities classification in DE's

    According to Mathworld, if in y''+P(x)y'+Q(x)y=0, P diverges at x=x_0 quicker than \frac{1}{(x-x_0)} or Q diverges at x=x_0 quicker than \frac{1}{(x-x_0)^2} then x_0 is called an essential singularity. What I don't understand is that let's suppose Q diverges like \frac{1}{(x-x_0)^5}. In that...
  31. T

    Singularities and Analyticity at z=0

    Homework Statement The Attempt at a Solution Both \displaystyle \frac{\cos(z)-1}{z^2} and \displaystyle \frac{\sinh(z)}{z^2} have 1 singular point at z=0. For (a): z=0 is a removable singularity since defining f(0)=1 makes it analytic at all z\in\mathbb{C}. z=0 is isolated...
  32. D

    Fourier transform and singularities

    Consider the Fourier transform of a complex function f(t): f(t)=\int_{-\infty}^\infty F(\omega)e^{-i\omega t} Here t and \omega are on real axis. Let's suppose f(t) is square integrable. Here are my questions: 1) Since f(t) is square integrable, so we have...
  33. MartinJH

    Are singularities shaped as a sphere.

    I haven't read much on black holes as its not something that interests me compared to other goings on. When ever I do read/hear about them I have always pictured them as being a flat, circular plane. From what I have read they form from collaspsing stars, I understand that much. So do they...
  34. S

    Solving integral equation with double ingegrals and singularities

    Hello I need help to solve the following integral equation: f(x,y,w)=137.03.*y.^2./((0.238.*exp(0.067.*y.^2)+1).*(w-5.26.*x.*y-2.63).*(w-5.26.*x.*y+2.63))=1+8478./(10828-w.^2-1.13.*j.*w) xmin=-1, xmax.=1, ymin=0, ymax=inf (nad can be taken 500 because the function decreases rapidly) I want...
  35. S

    Overcoming Singularity Issues in Numerical Integration with MATLAB

    Singularities Killing Me! Hey, I'm having singularity problems when integrating both of these equations using MATLAB. When I increase 'm' and 'n' to larger values I get these issues. I need to evaluate these equations for 'm' and 'n' as large as 2000. Anyone know how I can overcome...
  36. G

    Was Super String theory specifically designed to explain singularities?

    Was Super String theory (theory that attempts to unite General Relativity and the Standard Model of particle physics) specifically designed to explain singularities; ie: phenomena such as black holes and the big bang? Does all other phenomena obey, and can be explained, by either classical...
  37. K

    Classifying Singularities of f(z) in Complex Analysis

    f(z)=(z-1)((cos Pi z) / [(z+2)(2z-1)(z^2+1)^3(sin^2 Pi z)]
  38. A

    Find any singularities in the folloqing fucntion

    Find any singularities in the following function, say whether they are removable or non-removable. Indicate the limit of f(x) as x approaches the singularity. (x^(2) + x + 1) /( x-1) Not to sure where to start, as the numerator does not factorise easily.
  39. T

    Singularities, Density, and the Planck Length

    It is generally accepted that a star of sufficient mass collapsing in on itself will form a black hole (singularity) where density is infinite. I see a few problems arising with this, and I would like to have them clarified. 1.) Density=mass/area If the mass of any star is finite, how can an...
  40. Loren Booda

    Wrecks from racing track singularities

    Do most wrecks in automobile racing start at the singularities of the track - e.g., where the straight path becomes circular? Might this likewise be true of road driving?
  41. L

    Naked singularities: evidence against compactified dimensions?

    Hi, Here's a http://blogs.discovermagazine.com/cosmicvariance/2011/03/04/fractal-black-holes-on-strings/" ). Basically, they showed that if http://en.wikipedia.org/wiki/Black_string" is to be expected. However naked singularities would be such a strange thing that it seems safe to...
  42. FtlIsAwesome

    Naked singularities and traversable wormholes

    From what I know, for a wormhole to be traversable it must be held open by negative mass. A naked singularity is a black hole whose spin is enough that it counteracts its own gravity and allows the singularity itself to be seen. I had this thought: Could a wormhole composed of two naked...
  43. J

    Why do singularities mean that GR breaks down?

    Why do singularities mean that GR "breaks down?" The existence of singularities in the form of black holes as predicted by GR is universally accepted at this point. The fact that GR calculations don't work inside a black hole means only that we are incapable of penetrating the event horizon...
  44. B

    Complex Analysis Singularities and Poles

    Assume throughout that f is analytic, with a zero of order 42 at z=0. (a)What kind of zero does f' have at z=0? Why? (b)What kind of singularity does 1/f have at z=0? Why? (c)What kind of singularity does f'/f have at z=0? Why? for (a) I'm pretty sure it is a zero of order 41...
  45. Vorde

    Non-Gravitational Singularities

    I have just finished Black Holes and Time Warps by Kip Thorne as part of a high school reading class. Regardless of my personal thoughts of the book, I was captivated by the question, can a naked singularity exist? I was thinking along the following route (I guess I should mention that I...
  46. L

    Improper integrals with singularities at both endpoints.

    Homework Statement Study the continuity of \int\frac{dx}{x \sin x} from 0 to pi/2 That's 1/(xsinx), latex isn't showing up clearly for me. I've been having a go at simply solving it as an indefinite integral to evaluate it but I keep ending up with more complicated expressions if I try...
  47. J

    Coordinate singularities and coordinate transformations

    I have a metric of the form ds^2 = (1-r^2)dt^2 -\frac{1}{1-r^2}dr^2-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2 A singularity exists at r=\pm 1 . By calculating R^{abcd}R_{abcd} i found out that this singularity is a coordinate singularity. I found the geodesic equations for radial photons...
  48. J

    Locate and Classify Singularities

    Homework Statement Locate and classify the singularities of the following functions a) f(z) = 1 / (z^3*(z^2+1)) b) f(z) = (1 - e^z)/z c) f(z) = 1 / (1-e^z(^2)) d) f(z) = z / (e^(1/z)) Homework Equations The Attempt at a Solution I am not sure what I need to do when it asks me to locate...
  49. N

    Variable exponent causing number of singularities to change for residue?

    Homework Statement Determine the nature of the singularities of the following function and evaluate the residues. \frac{z^{-k}}{z+1} for 0 < k < 1 Homework Equations Residue theorem, Laurent expansions, etc. The Attempt at a Solution Ok this is a weird one since we've...
  50. N

    Evaluating Contour Integral w/ Multiple Singularities

    Contour integral with multiple singularities inside domain without residue theorem?? Homework Statement Evaluate \oint\frac{dz}{z^{2}-1} where C is the circle \left|z\right| = 2 Homework Equations Just learned contour integrals, so not much. Ok to use Cauchy's Integral formula (if...
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