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Math Help - complex analysis3

  1. #1
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    complex analysis3

    Suppose that f is an entire function, and there exists a nonnegative number \alpha such that
    |f(z)| <= \alpha|z|^n
    for all z in C. Show that f is a polynomial of degree at most n. (Hint: Use the Cauchy integral formula to show that f^(n+1)(w)=0 for all w in C.)

    Thank you!
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  2. #2
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    \alpha is meant to be theta.
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  3. #3
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    Quote Originally Posted by shaheen7 View Post
    Suppose that f is an entire function, and there exists a nonnegative number \alpha such that
    |f(z)| <= \alpha|z|^n
    for all z in C. Show that f is a polynomial of degree at most n. (Hint: Use the Cauchy integral formula to show that f^(n+1)(w)=0 for all w in C.)
    The Cauchy integral formula says that f^{(n+1)}(w) = \frac{(n+1)!}{2\pi i}\oint_C\frac{f(z)}{(z-w)^{n+2}}dz. The idea here is to take C to be a large circle, of radius R, centred at the origin. So make the substitution z = Re^{i\theta}, estimate the size of the integral and show that it can be made arbitrarily small if R is large enough.
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    using circle of radius R and using modulus f(z) <= alpha modulus z ^ n
    i got \int(f(z) / (z-w)^n+2) = (modulus f(z) * 2ipiR) / R^(n + 2) = alpha * R^n+1 *2ipi / R^n+2. Since R is in the denominator, do we get our result?
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  5. #5
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    Yes, that looks about right. It shows that by choosing R large enough, you can make f^{n+1}(w) arbitrarily small. But since f^{n+1}(w) is independent of R, it follows that it must be 0. If that holds for all w, then f has to be a polynomial of degree at most n.
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  6. #6
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    by f^{n+1}(w) is independent of R, do you mean to say we can take R to infinity?
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