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Essential Singularity

  1. Oct 26, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that if z0 is an isolated singularity of f(z) that is not removable, then z0 is an essential singularity for ef(z).


    2. Relevant equations
    z0 is a pole of f(z) of order N iff f(z) = g(z)/(z-z0)^N, where g is analytic at z0 and g(z0) is not 0, iff 1/f(z) is analytic at z0 and has a zero of order N, iff |f(z)| → ∞ as z → z0.

    Casorati-Weierstrass theorem

    3. The attempt at a solution1. The problem statement, all variables and given/known data
    I know what to do if z is an essential singularity. But I'm having trouble if z is a pole. It seems that what I must do is find a sequence of points going to z0 such that f(z) goes to infinity, and also a sequence of points going to z0 such that f(z) approaches some complex number. I don't know how to do that.
     
    Last edited: Oct 26, 2011
  2. jcsd
  3. Oct 27, 2011 #2
    You could rely on Picard's First Theorem: Any entire function that's not a polynomial has an essential singularity at infinity. So Exp(z) is an entire function and:

    [tex]Exp[\frac{A(z)}{(z-z_0)^n}] [/tex]

    therefore has an essential singularity at z=z0.
     
  4. Oct 27, 2011 #3
    I'm not supposed to know that theorem yet -- is there a way to prove it using only Casaroti-Weierstrass?
     
  5. Oct 27, 2011 #4
    Ok, how about this then: You'd agree that the function:

    [tex]e^{1/z}[/tex]

    has an essential singularity at zero right since the Laurent series is:

    [tex]e^{1/z}=\sum_{n=0}^{\infty} \frac{1}{z^n n!}[/tex]

    and the same would be true for:

    [tex]e^{1/z+1/z^2}[/tex]

    since you could multiply the individual sums for e^(1/z) and e^(1/z^2) and still end up with a series with an infinite number of terms with z in the denominator. And we could keep doing that to show that:

    [tex]e^{1/z+1/z^2+\cdots+1/z^n}[/tex]

    also has an essential singularity at zero. Ok then, so for simplicity assume f(z) has a pole at zero so we can write:

    [tex]f(z)=\frac{\phi(z)}{z^m}[/tex]

    with [itex]\phi(z)[/itex] analytic and not zero at zero so:

    [tex]\phi(z)=\sum_{k=0}^{\infty} a_k z^k[/tex]

    then:

    [tex]e^{f(z)}=\exp(\frac{\sum_{k=0}^{\infty} a_k z^k}{z^m})[/tex]

    [tex]=\exp(h(z)) \exp(\sum_{j=0}^{N}\frac{c_j}{z^j})[/tex]

    with h(z) now analytic and e^(h(z)) non-zero. What then can you say about the other part of that expression?
     
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