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Thread: Power Series

  1. #1
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    Power Series

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

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