Euler’s Formula and Taylor Series

Taylor Series

Taylor Series provide a way to express many functions f that are infinitely differentiable on some interval as a power series. If f is such a function, then the power series coefficients are given by Taylor’s Formula which says that if f(x)=a_0 + a_1x + a_2x^2 + a_3x^3 + ... , then a_n = \frac{f^{(n)}(0)}{n!}. This is discussed in Chapter 6 of Abbott’s text. Let’s use this idea and apply it to the functions sin(x) and cos(x).

Recall the derivatives of sin(x). In order they go cos(x), -sin(x), -cos(x), sin(x), ... . This cyclic behavior demonstrates that sin(x) is an infinitely differentiable function. In order to write the Taylor series centered at 0 for sin(x), the coefficients must be determined. Using the formula above, a_0 = \frac{f(0)}{0!} = \frac{sin(0)}{1} = 0 ; a_1 = \frac{f'(0)}{1!} = \frac{cos(0)}{1} = 1 ; a_2 = \frac{f''(0)}{2!} = \frac{-sin(0)}{2} = 0 ; a_3 = \frac{f^{3}(0)}{3!} = \frac{-cos(0)}{6} = \frac{-1}{6} . This will continue cyclically, giving us the general formula a_n = \begin{cases} 0 & n=even\\ \frac{1}{n!} & n= odd \end{cases} . Substituting these into the formula, we can obtain the power series for sin(x). It can be shown that sin(x) equals its power series everywhere, so sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... for all x\in \mathbb{R}.

Repeating this process for cos(x) gives a similar result. Notice that the cyclic order of derivatives following from cos(x) looks the same as sin(x), but is shifted by one. Because of this it makes sense for the Taylor Series expansion of cos(x) to look like the one above, but with the coefficients shifted by one. This means that the coefficients will be 0 when n is odd this time! Specifically, a_n = \begin{cases} 0 & n=odd\\ \frac{1}{n!} & n= even \end{cases} . Substituting these into the power series formula, we arrive at the series cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ....

Euler’s Formula

Taylor series have many applications in Mathematics, Physics, and other disciplines. One very important use of Taylor Series is present in the construction of Euler’s Formula. For anyone unfamiliar, Euler’s formula says that for any real number x, e^{ix} = cos(x) + isin(x). Recall that i is the complex number used to describe \sqrt{-1}. We can show that this is true using Taylor Series! (It turns out that much of the theory of power series evaluated at real numbers also holds for complex numbers.) It can be shown that e^{x} equals its Taylor series for all x\in \mathbb{C}. We will find the Taylor series of e^{ix} so we can compare it to the Taylor Series of cos(x) + isin(x) which we can determine from our work above.

Notice that \frac{d}{dx} e^{ix} = ie^{ix}. Repeating derivatives will continue to multiply the previous one by i. Similar to sin(x) and cos(x), we see that e^{ix} has cyclic derivatives. The first four are ie^{ix}; -e^{ix}; -ie^{ix}; e^{ix} and then it starts over. This shows that e^{ix} is an infinitely differentiable function. Using the Taylor Series formula for the coefficients, we get the function e^{ix} = 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} +\frac{x^4}{4!} + i\frac{x^5}{5!} - .... By grouping the terms containing i together, we get e^{ix} = (1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...) + i(x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...). The power series in the parentheses should look familiar! Crazy enough they are the Taylor Series expansion for cos(x) and sin(x) Therefore we can conclude that e^{ix} = cos(x) + isin(x) which is exactly what Euler’s Formula says!

Just by looking at the Taylor Series expansions of a few simple functions, we are able to come to a pretty amazing result! Not only is Euler’s Formula a neat example of Taylor Series, but it is actually used in many important calculations! The function e^{ix} is a solution for many partial differential equations and is therefore used heavily in many applications of Mathematics. Some examples are Electricity and Magnetism, Quantum Mechanics, and Topology! If nothing else I hope you have learned that Taylor Series do not just represent a few random theorems in our textbook, but are instead extremely powerful tools that are implemented in important and interesting scenarios in both Mathematics and Physics!

Posted on by Jeremy Strockoz
Categories: Fall 2016

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