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Math Help - pole or singularities

  1. #1
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    pole or singularities

    hi ,

    is exp z / sin z a pole or singularities or neither ??

    applying pole test,

    lim z-->0

    (z-0)( exp z / sin z)

    = z*expz / sin z

    sub 0 into equation, achieve 0 / 0,

    using hopital rule, i get ( z*exp z + exp z ) / cos z

    = 1 <--- can i conclued it's a pole ?


    If i have to apply hopital rule and achieve a constant, does that mean that it's always a RS ??

    many thanks ?
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  2. #2
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    Quote Originally Posted by Chris0724 View Post
    hi ,

    is exp z / sin z a pole or singularities or neither ??

    applying pole test,

    lim z-->0

    (z-0)( exp z / sin z)

    = z*expz / sin z

    sub 0 into equation, achieve 0 / 0,

    using hopital rule, i get ( z*exp z + exp z ) / cos z

    = 1 <--- can i conclued it's a pole ?


    If i have to apply hopital rule and achieve a constant, does that mean that it's always a RS ??

    many thanks ?
    Yes, z = 0 is a (simple) pole of f(z) = \frac{e^z}{\sin z}. The principle part of the Laurent series terminates after 1 term: Wolfram|Alpha
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Yes, z = 0 is a (simple) pole of f(z) = \frac{e^z}{\sin z}. The principle part of the Laurent series terminates after 1 term: Wolfram|Alpha
    hi mr fantastic,

    thanks a lot. but another question,

    if i were to apply L'hoptial rule, it doesn't necessary mean to be a RS , right ?

    For question asking the nature of singularties

    e.g

    sin z / z

    1st i try to use the pole test which gave me 0 which is not a pole but suspected to be a RS,

    2nd , sub in 0 to get 0 / 0 then use hoptial rule to achieve a 1.

    but for equation unable to get 0 / 0 is it conside as neither ?

    many thanks
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  4. #4
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    First, a pole is a "singularity". Did you mean an "essential singularity"? I don't know what you mean by "RS".

    A function is "analytic" at a point if it can be written as a power series with only non-negative powers. It has a "pole" at at point if it can be written as a power series with a finite number of negative powers. It has an "essential singularity" at a point if it can be only be written as a power series with an infinite number of negative powers.

    I have no idea what you mean by "applying L'Hopital's rule". Whether a point is a pole or an essential singularity will have a limit of infinity as you approach the point so "L'Hopital's rule" does not apply.
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    First, a pole is a "singularity". Did you mean an "essential singularity"? I don't know what you mean by "RS".

    A function is "analytic" at a point if it can be written as a power series with only non-negative powers. It has a "pole" at at point if it can be written as a power series with a finite number of negative powers. It has an "essential singularity" at a point if it can be only be written as a power series with an infinite number of negative powers.

    [snip]
    I'll add to this. RS probably means removable singularity.

    To complete the taxonomy begun by HoI: An isolated singular point z_0 of f(z) is removable if the Laurent series expansion of f(z) about z = z_0 has no principle part (that is, there are no negative powers).

    Alternatively, An isolated singular point z_0 of f(z) is removable if \lim_{z \rightarrow z_0} f(z) exists and is finite.

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