What is Singular points: Definition and 38 Discussions

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

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  1. V9999

    Book recommendations about singular points of algebraic curves

    I'm not quite sure if this is an appropriate question in this forum, but here is the situation. I have just finished my graduate studies. Now, I want to explore algebraic geometry. Precisely, I am interested in the following topics: Singular points of algebraic curves; General methods employed...
  2. Baibhab Bose

    A Understanding the Order of Poles in Complex Functions

    When The denominator is checked, the poles seem to be at Sin(πz²)=0, Which means πz²=nπ ⇒z=√n for (n=0,±1,±2...) but in the solution of this problem, it says that, for n=0 it would be simple pole since in the Laurent expansion of (z∕Sin(πz²)) about z=0 contains the highest negative power to be...
  3. F

    Stability of singular points in a discrete control system

    Homework Statement Give an example of a non-linear discrete-time system of the form x1(k + 1) = f1(x1(k), x2(k)) x2(k + 1) = f2(x1(k), x2(k)) With precisely four singular points, two of which are unstable, and two other singular points which are asymptotically stable. Homework Equations J =...
  4. D

    I Complex analysis - removable singular points

    Hi. I have 2 questions regarding removable singular points. 1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is...
  5. J

    Finding the singular points for this differential equation

    Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...
  6. W

    Series solution of ODE near singular points with trig

    Homework Statement Given the differential equation (\sin x)y'' + xy' + (x - \frac{1}{2})y = 0 a) Determine all the regular singular points of the equation b) Determine the indicial equation corresponding to each regular point c) Determine the form of the two linearly independent solutions...
  7. ion santra

    What is the Nature of Singularity in the Function f(x)=exp(-1/z)?

    what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number? now i arrive at two different results by progressing in two different ways. 1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity. 2) now again if i...
  8. Destroxia

    Series solution with regular singular points?

    1. Homework Statement ##x^{2}y'' + (x^{2} + 1/4)y=0## 3. The Attempt at a Solution First I found the limits of a and b, which came out to be values of a = 0, and b = 1/4 then I composed an equation to solve for the roots: ##r^{2} - r + 1/4 = 0## ##r=1/2## The roots didn't differ by an...
  9. RJLiberator

    Basic question on Determing Singular Points

    Determine the singular points of each function: f(z) = (z^3+i)/(z^2-3z+2) So it is my understanding that a singular point is one that makes the denominator 0 in this case. We see that (z-2)(z-1) is the denominator and we thus conclude that z =2, z=1 are singular points. f(z) =...
  10. V

    Big Bang: A True Singularity That is Coordinate Independent

    Consider a flat Robertson-Walker metric. When we say that there is a singularity at $$t=0$$ Clearly it is a coordinate dependent statement. So it is a "candidate" singularity. In principle there is "another coordinate system" in which the corresponding metric has no singularity as we...
  11. Soumalya

    TextBooks for Some Topics in Mathematics

    Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...
  12. I

    MHB Exploring Singular Points of a DE: w''+z*w'+kw=0

    Hi, I'm asked to find and classify the singular points of a function w(z) in the differential equation: w''+z*w'+kw=0 where k is some unknown constant. The only singular point I notice is z=\infty. Is that right? I did a transformation x=1/z and examined the singular point at x=0 and found...
  13. S

    Differential Equation -> Behaviour near these singular points

    Differential Equation ---> Behaviour near these singular points Homework Statement Problem & Questions: (a) Determine the two singular points x_1 < x_2 of the differential equation (x^2 – 4) y'' + (2 – x) y' + (x^2 + 4x + 4) y = 0 (b) Which of the following statements correctly describes...
  14. S

    Techniques for Solving Equations with Irregular Singular Points

    In our differential equations class, we learned about Ordinary and Regular Singular Points of a differential equation. We learned how to solve these equations with power series using the Frobenius method. I was wondering what happens when there is an irregular singular point, like...
  15. FeDeX_LaTeX

    Identifying Singular Points in the Equation of Motion: DE Homework

    Homework Statement The equation of motion of a particle moving in a straight line is ##x'' - x + 2x^3 = 0## and ##x = \frac{1}{\sqrt{2}}, x' = u > 0## at ##t = 0##. Identify the singular points in the phase plane and sketch the phase trajectories. Describe the possible motions of the...
  16. S

    Classifying Singular Points: Regular or Irregular?

    Homework Statement Find all singular points of xy"+(1-x)y'+xy=0 and determine whether each one is regular or irregular. Homework Equations The answer is x=0, regular. The Attempt at a Solution I know that x=0 since you set whatever is in front of y" to 0 and you solve for x, right...
  17. E

    Integration with branch cuts and singular points

    Homework Statement Prove that \int_0^{\infty} \frac{x^{1/\alpha}}{x^2-a^2} dx = \frac{\pi}{2a}\frac{a^{1/\alpha}}{\sin(\pi/a)}\left(1-\cos(\pi/\alpha)\right) where a>0 and -1<1/\alpha<1 Homework Equations It is apparent that there are two first order singular points at x=a and x=-a...
  18. J

    Finding singular points of a non-algebraic curve.

    Let F : \mathbb{R}^2 \rightarrow \mathbb{R}^2 be the map given by F(x, y) := (x^3 - xy, y^3 - xy). What are some singular points? Well, I know that for an algebraic curve, a point p_0 = (x_0, y_0) is a singular point if F_x(x_0, y_0) = 0 and F_y(x_0, y_0) = 0. However, this curve is not...
  19. N

    The singular points on f = x^2 y - x y on a plane

    Let f(x,y) = x^2 y - xy = x(x-1)y be a polynomial in k[x,y]. I am looking for the singular subset of this function. Taking the partials, we obtain f_x = 2xy - y f_y = x^2 - x. In order to find the singular subset, both partials (with respect to x and with respect to y) must vanish. So...
  20. P

    MHB Exploring Chebyshev Equation's Singular Points and Regularity

    I have computed the singular points of Chebyshev equation to be x= 1, -1. What is the best way to find whether they are regular? Thanks.
  21. J

    Wish to get singular points of algebraic functions

    Is there a numeric method to find singular points for managable algebraic functions? I have: w^2+2z^2w+z^4+z^2w^2+zw^3+1/4w^4+z^4w+z^3w^2-1/2 zw^4-1/2 w^5=0 and I wish to find the singular points for the function w(z). I can find them for simpler functions like w^3+2w^2z+z^2=0 In this...
  22. J

    Ordinary points, regular singular points and irregular singular points

    Say we have an ODE \frac{d^{2}x}{d^{2}y}+ p(x)\frac{dx}{dy}+q(x)y=0 Now, we introduce a point of interest x_{0} If p(x) and q(x) remain finite at at x_{0} is x_{0} considered as an ordinary point ? Now let's do some multiplication with x_{0} still being the point of interest...
  23. T

    Identifying and Classifying Singular Points in Differential Equations

    Homework Statement Locate the singular points of x^3(x-1)y'' - 2(x-1)y' + 3xy =0 and decide which, if any, are regular. The Attempt at a Solution In standard form the DE is y'' - \frac{2}{x^3} y' + \frac{3}{x^2(x-1)} y = 0. Are the singular points x=0,\pm 1\;? Regular singular...
  24. J

    Boundary, stationary, and singular points

    This is a topic in multi-variable calculus, extrema of functions. Our professor wrote: Boundary points: points on the edges of the domain if only such points stationary: points in the interior of the domain such that f is differentiable at x,y and gradient x,y is a zero vector...
  25. J

    Regular singular points of 2nd order ODE

    Homework Statement [PLAIN]http://img265.imageshack.us/img265/6778/complex.png I did the coefficient of the w' term. What about the w term? This seems like a fairly standard thing, but I can't seem to find it anywhere. What ansatz should I use for q, if the eqn is written w''+pw'+qw...
  26. J

    I can't see how to express those in simpler form.

    I need to find and classify the singular points and find the residue at each of these points for the following function; f(z) = \frac{z^{1/2}}{z^{2}+1} I can see that the singular points are at z=i and z=-i but have no idea how to classify them or find the residue at each point. I know...
  27. S

    Removing singular points

    Is there a correct mathematical procedure to remove singular points so as to create a smooth continuum, differentiable everywhere? For example,for cusp singularities, is some kind of acceptable "cutting and joining" procedure available at the limit? I asked a similar question in the topology...
  28. S

    Regular singular points (definition)

    Hello, I am trying to understand the definition of regular point, regular singular point and irregular point for example, the ode. what would be the r,rs or i points of this? x^3y'''(x)+3x^2y''(x)+4xy(x)=0 dividing gives the standard form y''+(3/x)y' + (4/x^2)y=0 So...
  29. F

    Why derive Regular Singular Points?

    I am currently studying a great text Elementary Differential Equations and Boundary Valued Problems 9th edition; and we have come to chapter 5 and are studying Ordinary Points, Singular Points, and Irregular Points. (get the point?) Anyway, I did see these mentioned,, this...
  30. S

    Transformations involving singular points

    Can a triangle be smoothly transformed to a circle?
  31. S

    Topological transform of singular points?

    Are singular points necessarily mapped to singular points under topological transformations? A specific example would a 2-space deformation of a triangle to any closed string with no cross over points. Would the three singular points of the triangle be necessarily mapped to three singular points...
  32. A

    Where Can I Find the Singular Points for a Pendulum System?

    hi, given the system ml^{2}\theta''+b\theta'+mglsin(\theta) how do I find the singular points?? or any system for that matter - trying the isocline method just not working! tedious..
  33. C

    Singular Points of the ODE: Identifying and Understanding

    Homework Statement For the ODE xy" + (2-x)y' + y = 0 i want to show it has one singular point and identify its nature Homework Equations The Attempt at a Solution I have read the topic and I see that a point Xo is called and ordinary point of the equation if both p(x) and...
  34. J

    Amount of singular points

    Is it possible for a complex analytic function to have an uncountable set of singular points?
  35. C

    Power Series & Singular Points: Why Change the Form?

    when finding a power series solution we have to put the differential equation ay''+by'+c=0 into the form y''+By+C=0 this leads to singular points when a=0 but why can't we leave the equation in its original form and use power series substitution to avoid singular points? or in...
  36. F

    Singular points in 3-dim space

    For a linearized system I have eigenvalues \lambda_1, \lambda_2 = a \pm bi \;(a>0) and \lambda_3 < 0 , then it should be an unstable spiral point. As t \to +\infty the trajectory will lie in the plane which is parallel with the plane spanned by eigenvectors v_1,v_2 corresponding to \lambda_1...
  37. O

    Classifying ordinary and singular points

    Hello, I am stuck on classifying the points with this DE...=\ xy''+(x-x^3)y'+(sin x)y=0 The solution says (sin x)/x is infinitely differentiable...so x=0 is an ordinary point? I was taught...if P(xo)=0, then xo is a singular point. Here P(x)=x...so x=0. So, what I don't get is the...
  38. L

    Solving 1+x^4=0: Finding the Singular Points

    I think I've got some minor braindamage or something but i just can't remember how to find the singular points of 1/(1+z^4) I guess the problem is to solve the equation 1+x^4=0 and get complex roots but this is what I don't remember how to do. Thanks.