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Math Help - Residue

  1. #1
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    Residue

    I have a doubt with the singularities and poles of a function, for example in the next complex function:

    f(x)=\sqrt{x^2+a}

    with a a constant. The function is zero when x=\pm{\sqrt{a}}

    For this value of x, is it a singularity or a pole? If it is a pole, of which order? Is it possible to calculate the residue for x=\pm{\sqrt{a}}?
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  2. #2
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    Quote Originally Posted by germana2006 View Post
    I have a doubt with the singularities and poles of a function, for example in the next complex function:

    f(x)=\sqrt{x^2+a}

    with a a constant. The function is zero when x=\pm{\sqrt{a}} Mr F says: No. You want {\color{red}x=\pm i {\sqrt{a}}}.

    For this value of x, is it a singularity or a pole? If it is a pole, of which order? Is it possible to calculate the residue for x=\pm{\sqrt{a}}?
    It's a branch point. A branch point is a special type of singularity.
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  3. #3
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    This may help

    f(x)=\sqrt{x^2+a}=e^{ln(\sqrt{(x^2+a)})}=e^{1/2ln(x^2+a)}
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    It's a branch point. A branch point is a special type of singularity.
    I disagree. To be a singularity we require that the function is analytic in a puntured neighborhood of that point. This is not true for \sqrt{z^2+a} along its branch.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    I disagree. To be a singularity we require that the function is analytic in a puntured neighborhood of that point. This is not true for \sqrt{z^2+a} along its branch.
    I quote from Churchill et al Complex Variables and Applications:

    "... a point z_0 is called a singular point of a function f if f faails to be analytic at z_0 but is analytic at some point in every neighbourhood of z_0. A singular point is said to be isolated if, in addition, there is some neighbourhood of z_0 throughout which f is analytic except at the point itself."

    A branch point is a singular point - what it's not is an isolated singular point.

    -----------------------------------------------------------------------------------------

    There are two categories of singularity:

    1. Isolated singularity.

    The point z = z_0 is an isolated singularity of f(z) if f(z) is not analytic at z = z_0 but is analytic in some deleted neighbourhood of z = z_0.

    There are three types:

    i. Removable. \lim_{z \rightarrow z_0} f(z) exists and is finite. Alternatively, the Laurent series expansion of f(z) about z = z_0 has no principle part.

    Eg. f(z) = \frac{\sin z}{z} has a removable singularity at z = 0.

    ii. Pole. (z - z_0)^m f(z) is analytic at z = z_0 for some positive integer m. The smallest such m is called the order of the pole. Alternatively, the principle part of the Laurent series expansion of f(z) about z = z_0 terminates after m terms.

    Eg. f(z) = \frac{1}{z-1} has a simple pole at z = 1.

    iii. Essential. It's neither removable nor a pole. Alternatively, the principle part of the Laurent series expansion of f(z) about z = z_0 does not terminate. Alternatively, \lim_{z \rightarrow z_0} f(z) does not exist (NB. Picard's Theorem).

    Eg. f(z) = e^{1/z} has an essential singularity at z = 0.

    2. Non-isolated singularity.

    If the singular point z = z_0 is not isolated, then it is a non-isolated singularity.

    Eg. The branch point z = 0 of f(z) = \ln z is a non-isolated singularity. In fact, any branch point of a multiple-valued function is a non-isolated singularity.
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  6. #6
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    Okay. I just never seen such terms before. But if you use that terminology then it is not just \pm i\sqrt{a} it then should be (-\infty,0].
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  7. #7
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    Quote Originally Posted by ThePerfectHacker View Post
    Okay. I just never seen such terms before. But if you use that terminology then it is not just \pm i\sqrt{a} it then should be (-\infty,0].
    Yes. But it should be noted that branch cuts are defined by convention ..... You say (-\infty,0], I could just as well define the branch cut to be [0, \infty) in which case the points in (-\infty,0] are not singular .....

    The branch point is (the only point that is) common to all branch cuts .... This makes it fundamentally different from all other points on a branch cut.
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