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Math Help - Power Series

  1. #1
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    Power Series

    Show that the power series  f(z) = 1+z+\frac{z^2}{2!}+ \cdots = \sum_{n=1}^{\infty} \frac{z^n}{n!} is equal to  e^z .

    So suppose  w \in \mathbb{C} . Then  f(z)f(w) =\sum_{n=1}^{\infty} \frac{z^n}{n!} \cdot \sum_{n=1}^{\infty} \frac{w^n}{n!} = f(z+w) . Also  f(x) = 1+x+\frac{x^2}{2!}+ \cdots = \sum_{n=1}^{\infty} \frac{x^n}{n!} . And so  f(x) = e^{x} . And then we need to show that  f(iy) = \cos y + i \sin y ?
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  2. #2
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    In other words we know that  \exp(wz) = \exp w + \exp z . So I established that in the first part by showing that  f(z) follows that property. So  e^z = e^{x+iy} = e^{x}e^{iy} . So I evaluated the real and imaginary parts of  f(z) .
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