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Math Help - Complex Singularities

  1. #1
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    Complex Singularities

    Bonjour!

    I have a few math questions involving locating and classifying singularities for the following functions:

    a) f(z) = \frac{1}{1-e^{z^{2}}}

    b) f(z) = \frac{z}{e^{\frac{1}{z}}}

    c) f(z) = \frac{1-e^{z}}{z}

    Merci!
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  2. #2
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    Hint: Singularities occur when the denominator is 0 + 0i.
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  3. #3
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    Quote Originally Posted by ComplexXavier View Post
    Bonjour!

    I have a few math questions involving locating and classifying singularities for the following functions:

    a) f(z) = \frac{1}{1-e^{z^{2}}}

    b) f(z) = \frac{z}{e^{\frac{1}{z}}}

    c) f(z) = \frac{1-e^{z}}{z}

    Merci!
    a) Poles of order 2 at values of z such that 1 - e^{z^2} = 0.

    b) Essential singularity at z = 0.

    c) Removable singularity at z = 0.

    I suggest you research the reasons why ....
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  4. #4
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    I understand b and c now and how we came about finding them.

    For a, I have found the singularities to be +/- sqrt(2*i*pi*n) but I am having trouble showing that these are poles of order 2. Any thoughts?
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  5. #5
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    What reason do you have to think that they are poles of order 2?
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  6. #6
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    Well, I tried to prove it to be removable or essentially and from my findings it was neither. So by process of elimination the only one I can't disprove is poles.
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  7. #7
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    Quote Originally Posted by ComplexXavier View Post
    I understand b and c now and how we came about finding them.

    For a, I have found the singularities to be +/- sqrt(2*i*pi*n) but I am having trouble showing that these are poles of order 2. Any thoughts?
    Can you show that z = 0 is a pole of order 2? There is a useful theorem in the second paragraph here: http://mathworld.wolfram.com/Pole.html (I'm sure the theorem is also in your class notes and textbook).
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