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Math Help - continous in the complex

  1. #1
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    continous in the complex

    How do i prove sin z is continuous in the complex region?

    Thanks Adam
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  2. #2
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    If you mean using the definition of continuity, the same way you would in any region- show that, for any complex number, z_0, given any \epsilon> 0, there exist a real number \delta> 0 so that if |z-z_0|< \delta, then |sin(z)- sin(z_0)|< \epsilon.

    Exactly how you would do that, depends on exactly how you are defining sin(z) for z complex. If you are defining sin(z) in terms of its Taylor series, just subtract the Taylor series for sin(z) and sin(z_0). If you defining sin(z) as \frac{e^{z)- e^{-z}}{2i}, are you allowed to use the fact that e^z is continuous?

    Or, since sin(z) satisfies the Cauchy-Riemann equations, it follows that it is differentiable and therefore continuous.
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  3. #3
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    If i do the Cauchy-Riemann equations does this imply its holomorphic?

    Or does it not work taht way around?
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