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Thread: 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
    $\displaystyle F(z)=e^{z^2}$

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

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

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

    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 $\displaystyle [0,z]$ represent the line joining $\displaystyle 0$ and $\displaystyle z$.

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