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Math Help - Complex An

  1. #1
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    Exclamation Complex An

    Hey I really need help with this questions, I honestly don't even know where to start...
    1)Explain why F(z)=e/\(z/\2) has an anti derivative in the whole plane?

    /\ is the exponent carrot thingy
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by confusedmathmajor View Post
    Hey I really need help with this questions, I honestly don't even know where to start...
    1)Explain why F(z)=e/\(z/\2) has an anti derivative in the whole plane?

    /\ is the exponent carrot thingy
    F(z)=e^{z^2}

    This is a composition of two functions that are analytic on \forall z \in \mathbb{C}

    let g(z)=e^z \mbox{ and } h(z)=z^2

    so F(z)=g(h(z))=e^{z^2}=\sum_{n=0}^{\infty}\frac{z^{2  n}}{n!}

    By the ratio test this converges for all values of z So we can integrate for the antiderivative.
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  3. #3
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    I like EmptySet's solution. Here is another. Let [0,z] represent the line joining 0 and z.

    Define, F(z) = \int_{[0,z]} e^{w^2} dw.
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