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Exponential Functions and Euler's Formula
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It is worth noting that the standard way to define the sine and cosine functions requires the concept of an angle, in degrees or radians, but no rigorous definition is provided This in fact is not easy to do and rarely discussed at all in calculus textbooks. Euler's formula now follows by setting $x = i\theta$ in \eqref{taylor} and splitting the series into its real and imaginary parts, \begin{align*} e^{i\theta} &= \sum_{k=0} \frac{(i\theta)^k}{k! The appendix provides a rigorous discussion showing how to use power series to solve a simple ordinary differential equation.
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