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Section 2.6 The 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}

Definition 2.4. 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:

\begin{equation} e^{z_1 + z_2}=e^{z_1}e^{z_2}\tag{2.6.3} \end{equation}

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

\begin{equation} e^z=e^{re^{i\theta}}.\tag{2.6.4} \end{equation}

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.

Figure 2.5. The graph of \(\Re e^z=\Re e^{x+iy}=e^x\cos y\text{.}\)