Definition 2.13. Analytic Continuation.
The process of extending the definition of a function from a real-valued argument to a complex-valued argument is called analytic continuation.
Section 2.10 Functions of a Complex Variable
In Section 2.9, we extended the definition of the exponential function to include complex numbers as inputs and in (2.6.3) and (2.6.4) we defined the sine and cosine of a pure imaginary argument. We can go further and extend the sine and cosine functions to be valid for a general complex-valued argument \(z\text{.}\) We define:
\begin{align}
\cos{z}\amp = \frac{1}{2}(e^{iz}+e^{-iz})\tag{2.10.1}\\
\sin{z}\amp = \frac{1}{2i}(e^{iz}-e^{-iz})\tag{2.10.2}
\end{align}
This process is done to make the extension as smooth (i.e. differentiable) as possible. You can learn more about analytic continuation in a course on complex-variable theory. This beautiful mathematics goes well beyond the current scope of this text.
The analytic continuations for powers and roots is discussed in Section 2.11 and for logarithms in Section 2.12
To Remember.
The good news is that any formulas that you memorized for real exponentials, real trigonometric functions, powers and roots, and logarithms also apply to their analytic continuations. In particular:
\begin{equation}
e^{z_1 + z_2}=e^{z_1}e^{z_2}\tag{2.10.3}
\end{equation}
Also, the analytic continuations of these common functions of a real variable are unique.
