Why can we plug complex numbers into maclaurin series

algebra-precalculuscomplex numberscomplex-analysislogictaylor expansion

When finding a Maclaurin series for a function f(x). We evaluate f '(0), f ''(0), etc, to find our coefficients for each term.
I have done this for the standard functions, $f (\mathbb{R}):->\mathbb{R}$. However, to prove $e^{ix} = cosx + i sinx$, one can substitute ix into the Maclaurin series for $e^x$. So I am trying to understand how did we extend the domain of our function, and how can I possibly do the same with other functions, e.g. $sinx$ so that we can have $x \in \mathbb{C}$.

Best Answer

What is typically done is some variation of the following:

  1. We define limits of complex functions similarly to their real counterparts but using the modulus in place of the absolute value - so $\lim_{z \rightarrow z_0} f(z) = L$ if $|f(z) - L| \rightarrow 0$ as $|z - z_0| \rightarrow 0$, where the $|\cdot|$ represent complex modulus operations. We do something similar for the complex derivative.

  2. We define $e^z = \sum_{n = 0}^\infty \frac{z^n}{n!}$, and we note that this gives us an entire function (i.e. it is analytic for all $z \in \mathbb{C}$) that aligns with the version of the function defined for real numbers. Again, all of the results from Taylor's theorem have complex analogues, which is why this series is absolutely convergent.

  3. We then define $\cos z = \frac{e^{iz} + e^{iz}}{2}$ and $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$.

  4. We can then note that because $e^z$ is entire, we can directly derive the Taylor series for $\cos z$ and $\sin z$ and we get the versions that line up with the real versions, and also it's very easy to see that $e^{iz} = \cos z + i \sin z$.

The part where we see that the functions we've defined happen to line up with known functions on the real line is known as analytic continuation, although it's usually presented a little differently (we usually start with a function defined on the real numbers, find its Taylor series, and then look at a region of complex numbers where that series converges and say that the series defines an analytic continuation of our function in that region).

Related Question