Definition 3.4.1.
The (complex) exponential function is \(\exp: \C \to \C\text{,}\) defined for \(z=x+iy\) as
\begin{equation*}
\exp(z) \ := \ e^x \left( \cos y + i \sin y \right) \ = \ e^x e^{iy}\text{.}
\end{equation*}
Section 3.4 Exponential and Trigonometric Functions
To define the complex exponential function, we once more borrow concepts from calculus, namely the real exponential function and the real sine and cosine, and we finally make sense of the notation \(e^{it} = \cos t + i
\sin t\text{.}\)
1
How to define the real exponential function is a nontrivial question. Our preferred way to do this is through a power series: \(e^x = \sum_{k \geq 0} \frac 1 {k!} x^k\text{.}\) In light of this definition, you might think we should have simply defined the complex exponential function through a complex power series. In fact, this is possible (and an elegant definition); however, one of the promises of this book is to introduce complex power series as late as possible. We agree with those readers who think that we are cheating at this point, as we borrow the concept of a (real) power series to define the real exponential function.
This definition seems a bit arbitrary. Our first justification is that all exponential rules that we are used to from real numbers carry over to the complex case. They mainly follow from Proposition 1.2.7 and are collected in the following.
Proposition 3.4.2.
For all \(z, z_1, z_2 \in \C\text{,}\)
-
\(\displaystyle \exp \left( z_1 \right) \exp \left( z_2 \right) = \exp \left( z_1 + z_2 \right)\)
-
\(\displaystyle \frac 1 {\exp \left( z \right)} = \exp \left( -z \right)\)
-
\(\displaystyle \exp \left( z + 2 \pi i \right) = \exp \left( z \right)\)
-
\(\displaystyle \left| \exp \left( z \right) \right| = \exp \left( \Re z \right)\)
-
\(\displaystyle \exp(z) \ne 0\)
-
\(\frac{ d }{ d z } \exp \left( z \right) = \exp \left( z \right) \, \text{.}\)
Item (c) is very special and has no counterpart for the real exponential function. It says that the complex exponential function is periodic with period \(2 \pi i\text{.}\) This has many interesting consequences; one that may not seem too pleasant at first sight is the fact that the complex exponential function is not one-to-one.
Item (f) is not only remarkable, but we invite you to meditate on its proof below; it gives a strong indication that our definition of \(\exp\) is reasonable. We also note that (f) implies that \(\exp\) is entire.
Proof.
(f) The partial derivatives of \(f(z) = \exp(z)\) are
\begin{equation*}
\fderiv fx \ = \ e^x \left( \cos y + i \sin y \right) \qquad
\text{ and } \qquad \fderiv fy \ = \ e^x \left( -\sin y + i \cos
y \right) \text{.}
\end{equation*}
They are continuous in \(\C\) and satisfy the Cauchy–Riemann equation (2.2):
\begin{equation*}
\fderiv fx(z) \ = \ -i \, \fderiv fy(z)
\end{equation*}
for all \(z \in \C\text{.}\) Thus Theorem 2.3.1 says that \(f(z) = \exp(z)\) is entire with derivative
\begin{equation*}
f'(z) \ = \ \fderiv fx(z) \ = \ \exp(z) \, \text{.}
\end{equation*}
We should make sure that the complex exponential function specializes to the real exponential function for real arguments: indeed, if \(z = x \in \R\) then
\begin{equation*}
\exp(x) \ = \ e^x \left( \cos 0 + i \sin 0 \right) \ = \ e^x\text{.}
\end{equation*}
The trigonometric functions—sine, cosine, tangent, cotangent, etc.—also have complex analogues; however, they do not play the same prominent role as in the real case. In fact, we can define them as merely being special combinations of the exponential function.
Definition 3.4.4.
The (complex) sine and cosine are defined as
\begin{align*}
\sin z \amp \ := \ \tfrac{1}{2i} \left( \exp (iz) - \exp
(-iz) \right)\\
\cos z \amp \ := \ \tfrac 1 2 \left( \exp (iz) + \exp (-iz)
\right) \,\text{,}
\end{align*}
respectively. The tangent and cotangent are defined as
\begin{align*}
\tan z \amp \ := \ \frac{ \sin z }{ \cos z } \ = \ -i \, \frac{
\exp (2iz) - 1 }{ \exp (2iz) + 1 }\\
\cot z \amp \ := \ \frac{ \cos z }{ \sin z } \ = \ i \,
\frac{ \exp (2iz) + 1 }{ \exp (2iz) - 1 } \,\text{,}
\end{align*}
respectively.
Note that to write tangent and cotangent in terms of the exponential function, we used the fact that \(\exp(z) \exp(-z) = \exp(0) = 1\text{.}\) Because \(\exp\) is entire, so are \(\sin\) and \(\cos\text{.}\)
As with the exponential function, we should make sure that we’re not redefining the real sine and cosine: if \(z = x \in \R\) then
\begin{align*}
\sin z \amp \ = \ \tfrac{1}{2i} \left( \exp (ix) - \exp (-ix)
\right)\\
\amp \ = \ \tfrac{1}{2i} \left( \cos x + i \sin x - \cos(-x) - i
\sin(-x) \right)\\
\amp \ = \ \sin x\text{.}
\end{align*}
A similar calculation holds for the cosine. Not too surprisingly, the following properties follow mostly from Proposition 3.4.2.
Proposition 3.4.5.
For all \(z, z_1, z_2 \in \C\text{,}\)
\begin{align*}
\sin (-z) \amp = - \sin z\\
\cos (-z) \amp = \cos z\\
\sin (z + 2 \pi) \amp = \sin z\\
\cos (z + 2 \pi) \amp = \cos z\\
\tan (z + \pi) \amp = \tan z\\
\cot (z + \pi) \amp = \cot z\\
\sin (z + \tfrac \pi 2) \amp = \cos z\\
\cos (z + \tfrac \pi 2) \amp = -\sin z\\
\sin \left( z_1 + z_2 \right) \amp = \sin z_1 \cos z_2 +
\cos z_1 \sin z_2 \\
\cos \left( z_1 + z_2 \right) \amp = \cos z_1 \cos z_2 -
\sin z_1 \sin z_2\\
\cos^2 z + \sin^2 z \amp = 1\\
\cos^2 z-\sin^2 z \amp = \cos(2z)\\
\frac{ d }{ \diff{z} } \sin z \amp = \cos z\\
\frac{ d }{ \diff{z} } \cos z \amp = - \sin z \,\text{.}
\end{align*}
Finally, one word of caution: unlike in the real case, the complex sine and cosine are not bounded—consider, for example, \(\sin (iy)\) as \(y \to \pm \infty\text{.}\)
We end this section with a remark on hyperbolic trig functions. The hyperbolic sine, cosine, tangent, and cotangent are defined as in the real case:
Definition 3.4.6.
\begin{align*}
\sinh z \amp \ = \ \tfrac 1 {2} \left( \exp (z) - \exp (-z)
\right)\\
\cosh z \amp \ = \ \tfrac 1 2 \left( \exp (z) + \exp (-z) \right)\\
\tanh z \amp \ = \ \frac{ \sinh z }{ \cosh z } \ = \ \frac{ \exp
(2z) - 1 }{ \exp (2z) + 1 }\\
\coth z \amp \ = \ \frac{ \cosh z }{ \sinh z } \ = \ \frac{
\exp (2z) + 1 }{ \exp (2z) - 1 } \,\text{.}
\end{align*}
As such, they are yet more special combinations of the exponential function. They still satisfy the identities you already know, e.g.,
\begin{equation*}
\frac{d}{\diff{z}} \sinh z \ = \ \cosh z \qquad \text{ and }
\qquad \frac{d}{\diff{z}} \cosh z \ = \ \sinh z \, \text{.}
\end{equation*}
Moreover, they are related to the trigonometric functions via
\begin{equation*}
\sinh (iz) \ = \ i \sin z \qquad \text{ and } \qquad \cosh (iz) \ = \
\cos z \, \text{.}
\end{equation*}
