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Math Help - Need some compex analysis help

  1. #1
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    Need some compex analysis help

    I could use some help with this problem:

    Suppose that f is analytic on a domain D and has a zero of order m at z(subscript 0) in D. Show that:

    1.) f' has a zero of order m - 1 at z(subscript 0), and
    2.) f^2 has a zero of order 2m at z(subscript 0)
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  2. #2
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    Quote Originally Posted by spoon737 View Post
    I could use some help with this problem:

    Suppose that f is analytic on a domain D and has a zero of order m at z(subscript 0) in D. Show that:

    1.) f' has a zero of order m - 1 at z(subscript 0))
    Here is an attempt.

    We can write,
    f(z) = (z-z_0)^m * g(z) for all z in D.

    Then,

    f'(z)=m(z-z_0)^{m-1} *g(z) + (z-z_0)^m*g'(z)
    f'(z)=(z-z_0)^{m-1}*[mg(z)+(z-z_0)*g'(z)]

    We need to show that the second function part does not attain a zero at z_0. Indeed! Substitute and see:
    mg(z_0)+0*g'(z_0)=mg(z_0)!=0
    Because m!=0 and g(z_0)!=0.

    Thus, all the zero's belong to the first factor (z-z_0)^{m-1}.
    Which is of order m-1.
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