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Basic question on Determing Singular Points

  1. Jun 25, 2015 #1

    RJLiberator

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    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) = (2z+1)/(z(z^2+1))

    So, z=0, +/- i are singular points.

    Am I understanding this correctly?
     
  2. jcsd
  3. Jun 25, 2015 #2

    RUber

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    That is how I understand it. In general if you can find a value of z for which f(z) is undefined that is a singularity. You should also check what happens in the numerator to be sure you aren't missing anything.
     
  4. Jun 25, 2015 #3

    RUber

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    The formal definition for isolated singularities (or singular points), as stated in Fischer's "Complex Variables" is:
    "An analytic function f has an isolated singularity at a point z_0 if f is analytic in the punctured disc 0<|z-z_0|<r, for some r>0.
    That is, the function is well-defined in the neighborhood of the point, but not at the point itself.
     
  5. Jun 26, 2015 #4

    HallsofIvy

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    Given that f(z) is a rational function, a polynomial divided by a polynomial, then all singular points are where the denominator is 0.
     
  6. Jun 26, 2015 #5

    RJLiberator

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    Excellent. Thank you for this confirmation.

    This class is moving fasttttt.

    =)
     
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