## Section2.6The Exponential Function

It turns out that equations (2.5.3) and (2.5.4) for the sine and cosine of a pure imaginary argument can be extended to arbitrary complex values of the argument. For a general complex-valued argument $$z\text{,}$$ we define:

\begin{align} \cos{z}\amp = \frac{1}{2}(e^{iz}+e^{-iz})\tag{2.6.1}\\ \sin{z}\amp = \frac{1}{2i}(e^{iz}-e^{-iz})\tag{2.6.2} \end{align}

### Definition2.4.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.

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. The good news is that any formulas that you memorized for real exponentials and real trigonometric functions also apply to their analytic continuations. In particular:

$$e^{z_1 + z_2}=e^{z_1}e^{z_2}\tag{2.6.3}$$

In practice, if you need to find the real and imaginary parts of a complex-valued function of the complex variable $$z\text{,}$$ it will be easiest if you express $$z$$ in rectangular form, $$z=x+iy\text{.}$$ If you use the exponential form $$z=re^{i\theta}$$ you can end up having to evaluate exponentials of exponentials

$$e^z=e^{re^{i\theta}}.\tag{2.6.4}$$

The right-hand graph on the applet below shows the graph of the real part of the complex-valued exponential $$\Re e^z=\Re e^{x+iy}=e^x\cos y\text{.}$$ The slider choses various cross-sections of $$\Re e^z$$ which range from the real-valued exponential function $$\Re e^x$$ when $$y=0$$ to the real-valued trigonmetric function $$\cos y$$ when $$x=0\text{.}$$ The left-hand graph shows just a graph of the cross-section. Notice the damped (for negative values of $$x$$) cosine function for cross-sections between the two limiting cases.