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Math Help - Laurent expansion trouble!

  1. #1
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    Laurent expansion trouble!

    I need to find the Laurent expansion about 0 for the function:

    f(z)=exp(-1/z^4)

    I've done Laurent expansions before for functions involving exp(z), and used the taylor series to expand exp(z), but I'm stumped on what to do for this question. I havn't come across any other examples like this..

    Please help!
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  2. #2
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    Quote Originally Posted by gizmo View Post
    I need to find the Laurent expansion about 0 for the function:

    f(z)=exp(-1/z^4)

    I've done Laurent expansions before for functions involving exp(z), and used the taylor series to expand exp(z), but I'm stumped on what to do for this question. I havn't come across any other examples like this..

    Please help!
    Replace z with -1/z^4 in the usual Maclaurin series for e^z.
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  3. #3
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    that seems blindingly obvious now... sorry :/
    thanks for the speedy reply!
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  4. #4
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    ok so far my working is as follows:

    f(z)=exp(-1/z^{-4})

    [replace  z with -1/z^{-4} in power series for exponential function, to obtain Laurent series]

    f(z)=\sum (-1^n)(z^{-4n})/n!

    [as N \rightarrowinfinity , the approximation becomes exact for all complex numbers z, except at z=0]

    f(z)=1-(z^{-4})+(z^{-8})/2!-(z^{-12})/3!+...

    I have now written out 4 non-zero terms for this question and the function, is what I've done correct? Would you add anything in?

    Thanks. (I couldn't get the z^-4 to display correctly, sorry for the poor coding!)
    Last edited by mr fantastic; April 1st 2010 at 03:59 PM. Reason: Fixed latex.
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  5. #5
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    Quote Originally Posted by gizmo View Post
    ok so far my working is as follows:

    f(z)=exp(-1/z^{-4})

    [replace  z with -1/z^{-4} in power series for exponential function, to obtain Laurent series]

    f(z)=\sum (-1^n)(z^{-4n})/n!

    [as N \rightarrowinfinity , the approximation becomes exact for all complex numbers z, except at z=0]

    f(z)=1-(z^{-4})+(z^{-8})/2!-(z^{-12})/3!+...

    I have now written out 4 non-zero terms for this question and the function, is what I've done correct? Would you add anything in?

    Thanks. (I couldn't get the z^-4 to display correctly, sorry for the poor coding!)
    Yes.
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