The Complex Exponential Function

When the world says, ‘Give up,’ hope whispers, ‘Try it one more time,’ Lyndon B. Johnson

Analytic functions

Definition. A complex function f is said to be analytic analytic (or holomorphic) at a point $z_0$ if it satisfies any of the following equivalent conditions:

1. Differentiable in a neighborhood. There exists an open neighborhood U of $z_0$ such that f is complex differentiable at every point in U. Complex differentiability means that the derivative $f'(z)$ exists for all $z \in U$ in the complex sense: $f'(z) = \lim_{h \to 0} \frac{f(z+h)-f(z)}{h}.$

   Here the limit must be the same regardless of the direction from which h approaches zero in the complex plane. This is the crucial distinction from real differentiability: the increment $h$ is a complex number that can approach zero along any path — horizontal, vertical, spiral, or otherwise.

2. Open disc criterion. There exists some radius r > 0 such that f is complex differentiable at every point in the open disc B(z; r) = $\\{z : |z - z_0| < r\\}$. This is a specific case of Condition 1, since an open disc is just a particular or specific type of open neighborhood around $z_0$.
3. Power series expansion. The function can be locally expressed as a convergent power series:$f(z) = \sum_{n = 0}^\infty a_n(z -z₀)^n$ in some neighborhood of $z_0$. This series converges absolutely and uniformly on compact subsets within its disk of convergence.

The Complex Exponential Function

The complex exponential function, denoted as $e^z$ or $\exp(z)$, is one of the most important functions in all of mathematics. It is the unique entire function that extends the real exponential function $e^x$ to the complex plane. This function is not only entire but also periodic.

Construction and Definition

To define the complex exponential function, we seek a function $f: \mathbb{C} \to \mathbb{C}$ that satisfies two key properties:

1. Additivity (Functional Equation): $f(z_1 + z_2) = f(z_1) \cdot f(z_2)$ for all $z_1, z_2 \in \mathbb{C}$.
2. Consistency with the Real Exponential: $f(x) = e^x$ for all $x \in \mathbb{R}$.

Derivation

Let $z = x + iy$ with $x, y \in \mathbb{R}$. Using properties (1) and (2): $f(z) = f(x + iy) \overset{(1)}{=} f(x) \cdot f(iy) \overset{(2)}{=} e^x \cdot f(iy).$

Writing $f(iy) = A(y) + iB(y)$ where $A$ and $B$ are real-valued functions of $y$, $f(z) = e^x A(y) + ie^x B(y),$ so $u(x, y) = e^x A(y)$ and $v(x, y) = e^x B(y)$.

For $f$ to be analytic (differentiable everywhere), its real and imaginary parts must satisfy the Cauchy–Riemann equations.

Computing partial derivatives: $u_x = e^x A(y), \quad u_y = e^x A'(y), \quad v_x = e^x B(y), \quad v_y = e^x B'(y).$

The equation $u_x = v_y$ gives $e^x A(y) = e^x B'(y)$, hence $A(y) = B'(y) (\star)$.

Similarly, the equation $u_y = -v_x$ gives $e^x A'(y) = -e^x B(y)$, hence $A'(y) = -B(y) (\star\star)$.

Solving the Resulting ODE

Differentiating $A(y) = B'(y) (\star)$, we get $B''(y) = A'(y)$ and substituting $A'(y) = -B(y) (\star\star)$: $B''(y) = A'(y) = -B(y) \quad \Longrightarrow \quad B''(y) + B(y) = 0.$

This is a linear, homogeneous, constant-coefficient ODE with characteristic equation $r^2 + 1 = 0$, giving roots $r = \pm i$.

Recall. For complex conjugate roots $a \pm bi$, the general solution is $y(t) = e^{at}(\alpha\cos(bt) + \beta\sin(bt))$, where $\alpha, \beta \in \mathbb{R}$.

With $a = 0$ and $b = 1$ ($r = \pm i$), the general solution is: $B(y) = \alpha\cos y + \beta\sin y, \qquad A(y) =[(\star)] B'(y) = -\alpha\sin y + \beta\cos y.$

Determining the Constants

From property (2), $f(0) = e^0 = 1$. But $f(0) = e^0 A(0) + ie^0 B(0) = A(0) + iB(0)$, so $A(0) = 1 \quad \text{and} \quad B(0) = 0.$

From $A(0) = 1$: $-\alpha\sin 0 + \beta\cos 0 = \beta = 1$.

From $B(0) = 0$: $\alpha\cos 0 + \beta\sin 0 = \alpha = 0$.

Therefore $A(y) = \cos y$ and $B(y) = \sin y$, and we finally arrive at the definition of the complex exponential function: $\boxed{e^z = e^x(\cos y + i\sin y), \qquad z = x + iy.}$

Properties

• Well-Definedness and Analyticity. The function $e^z = e^x(\cos y + i\sin y)$ is well-defined for all $z \in \mathbb{C}$. It is an entire function (analytic on all of $\mathbb{C}$), which can be verified in two independent ways:

1. Direct verification via the Cauchy–Riemann equations. With $u(x, y) = e^x\cos y$ and $v(x, y) = e^x\sin y$: $\frac{\partial u}{\partial x} = e^x\cos y = \frac{\partial v}{\partial y} \;\checkmark, \qquad \frac{\partial u}{\partial y} = -e^x\sin y = -\frac{\partial v}{\partial x} \;\checkmark.$
   All partial derivatives are continuous everywhere, so by the sufficient condition for complex differentiability, $e^z$ is analytic at every point.
2. Power series representation. The exponential function admits the globally convergent power series:
   $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}.$
   This is a power series centered at $z_0 = 0$ with coefficients $a_n = 1/n!$. To determine its radius of convergence, we could apply the ratio test:
   $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n!}{(n+1)!} = \lim_{n \to \infty} \frac{1}{n+1} = 0 < 1.$
   Since $L = 0$, the radius of convergence is $R = 1/L = \infty$. The series converges absolutely for every $z \in \mathbb{C}$, confirming that $e^z$ is entire.
   Recall: Any function defined by a power series with infinite radius of convergence is entire.
   A power series centered at a has the form $f(z)=\sum _{n=0}^{\infty }c_n(z-a)^n$. A classical theorem from complex analysis states that a power series converges and defines a holomorphic function on the open disk $|z - a| < R$, where R is its radius of convergence. Inside that disk, the function is infinitely differentiable and its derivatives are given by term‑by‑term differentiation.

• Addition Formula: $e^{z_1 + z_2} = e^{z_1} \cdot e^{z_2} \quad \text{for all } z_1, z_2 \in \mathbb{C}.$

Write $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. Then,

$$ \begin{aligned} e^{z_1}⋅e^{z_2} = &[\text{By definition}] e^{x_1}(cos(y_1) + isin(y_1))⋅e^{x_2}(cos(y_2) + isin(y_2)) \\[2pt] &= e^{x_1 + x_2}\bigl[(\cos y_1\cos y_2 - \sin y_1\sin y_2) + i(\sin y_1\cos y_2 + \cos y_1\sin y_2)\bigr]. \\[2pt] &[\text{Using the angle addition formula for cosine and sine}] = e^{x_1 + x_2}(\cos(y_1 + y_2) + i\sin(y_1 + y_2)) \\[2pt] & = e^{(x_1 + x_2) + i(y_1 + y_2)} = e^{z_1 + z_2}. \qquad \blacksquare \end{aligned} $$

• Consistency with the Real Exponential If $z$ is real (i.e., $y = 0$): $\exp(z) = \exp(x + i \cdot 0) = e^x(\cos 0 + i\sin 0) = e^x.$

The complex exponential restricted to $\mathbb{R}$ recovers the familiar real exponential function.

• Modulus and Argument. $|e^z| = |e^x(\cos y + i\sin y)| = e^x\sqrt{\cos^2 y + \sin^2 y} = e^x.$ The argument of $e^z$ is any angle whose cosine and sine are $\cos y$ and $\sin y$ respectively. Because cosine and sine are periodic with period $2\pi$, every number of the form $y + 2k\pi$ ($k \in \mathbb{Z}$) is a valid argument: $\arg(e^z) = y + 2k\pi, k \in \mathbb{Z}.$

This shows that the real part of z controls the magnitude of $e^z$, while the imaginary part controls its direction (angle).

• Non-Vanishing Property. Since $|e^z| = e^x > 0$ for all $x \in \mathbb{R}$, it follows that $e^z \neq 0 \quad \text{for all } z \in \mathbb{C}.$

The complex exponential never takes the value zero. This is a fundamental distinction from polynomial functions, which (by the Fundamental Theorem of Algebra) always have zeros.

• Periodicity: The complex exponential function exhibits periodic behavior with a fundamental period of $2\pi i$. $e^{z+2k\pi·i} = e^{z}e^{2k\pi·i} =[\text{Euler's formula}] e^{z}(cos(2k\pi·i)+isin(2k\pi·i)) = e^{z}(1 + i·0) = e^{z}$ for every integer $k$. This means $\exp$ takes the same value at $z$, $z + 2\pi i$, $z + 4\pi i$, and so on and so forth. The real exponential $e^x$ is strictly monotone and injective; the complex exponential, by contrast, is infinitely-to-one: every value in its range is attained at infinitely many points, arranged in vertical columns spaced $2\pi$ apart.

The periodicity of $e^z$ is the fundamental reason why the complex logarithm is multi-valued. If $e^{z_0} = w$, then $e^{z_0 + 2k\pi i} = w$ for every integer $k$, so “$\log w$” has infinitely many possible values.

• Solutions to $e^z = \alpha$. For any non-zero complex number $\alpha$, the equation $e^z = \alpha$ has infinitely many solutions.

Writing $\alpha$ in polar form as $\alpha = Re^{i\theta}$ where $R = |\alpha| > 0$ and $\theta = \arg(\alpha)$: $e^z = e^{x+iy} = Re^{i\theta} \quad \Longrightarrow \quad e^x = R, y = \theta + 2k\pi$. Thus, $x = \ln(R)$ and the solutions are: $\boxed{z = \ln(R) + i(\theta + 2k\pi)}, \quad k \in \mathbb{Z}.$

This is effectively the definition of the (multi-valued) complex logarithm $\log(\alpha)$. The solutions form an infinite set of points equally spaced along a vertical line at $x = \ln|\alpha|$, with vertical spacing $2\pi$.

Example. Solve $e^z = 2 + 2i$.

1. Express $2 + 2i$ in polar form: $|2 + 2i| = 2\sqrt{2}, \qquad \arg(2 + 2i) = \frac{\pi}{4}.$ So $2 + 2i = 2\sqrt{2}\,e^{i\pi/4}$.
2. The solutions are $z = \ln(2\sqrt{2}) + i\!\left(\frac{\pi}{4} + 2k\pi\right) = \frac{3}{2}\ln 2 + i\!\left(\frac{\pi}{4} + 2k\pi\right), \qquad k \in \mathbb{Z}.$
3. There are infinitely many solutions, reflecting the $2\pi i$-periodicity of $e^z$.

• The exponential is its own derivative. The derivative of the exponential function satisfies the remarkable identity: $\frac{d}{dz} e^z = e^z.$

Proof via the Cauchy–Riemann equations. Since $e^z = e^x(cos(y) + isin(y)) = u + iv$, $u = e^x\cos y$ and $v = e^x\sin y$:

$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = e^x\cos y + ie^x\sin y = e^z. \qquad \blacksquare$

Proof via Power series.

Theorem. If a function $f(z)$ can be represented by a power series $f(z) = \sum_{n=0}^{\infty} a_n(z - z_0)^n$ that converges in a disk $|z - z_0| < R$ (with $R > 0$), then $f$ is analytic within that disk, and its derivative is obtained by differentiating term by term: $f'(z) = \sum_{n=1}^{\infty} n\,a_n(z - z_0)^{n-1}.$

Applying this to $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ (centered at $z_0 = 0$ with radius of convergence $R = \infty$):

$\frac{d}{dz} e^z = \sum_{n=1}^{\infty} \frac{n z^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{z^{n-1}}{(n-1)!} \overset{k = n-1}{=} \sum_{k=0}^{\infty} \frac{z^k}{k!} = e^z. \qquad \blacksquare$

Since $(e^z)' = e^z$ and $e^z$ is entire, repeated differentiation yields the same result $(e^z)'' = e^z$, $(e^z)''' = e^z$, and so on — all derivatives of all orders exist and equal $e^z$ itself. This confirms that the complex exponential models processes where the rate of change is proportional to the current state, just like its real counterpart.

• $e^z$ is the unique solution to the first-order initial value problem $\frac{df}{dz} = f(z)$ with initial condition f(0) = 1.

The given differential equation is a classic example of a separable differential equation. The equation can be rewritten as $\frac{df}{f} = dz$. Integrating both sides gives: $\int \frac{df}{f} = \int dz \leadsto ln|f| = z + C$

To solve for f, we exponentiate both sides: $|f| = e^{z + C} = e^{z}e^{C}$. Let’s rename the constant $e^{C}$ as A. Since $e^{C}$ is always positive, A can be any non-zero real number. The general solution is: $\boxed{f(z) = Ae^z}$ where $A = e^C$ is an arbitrary nonzero constant.

Now, we use the initial condition f(0) = 1 to find the value of A: $f(0) = A \cdot e^0 = A = 1$. Plugging this value back into the general solution gives us the specific solution: $f(z) = Ae^z = 1e^z = e^z$. This is the unique solution to the initial value problem. This characterization provides yet another way to define the exponential function: it is the unique analytic function satisfying $f' = f$ and $f(0) = 1$.

• Euler's formula. Setting $x = 0$ in the definition of the exponential function $e^z = e^x(\cos y + i\sin y)$ yields one of the most famous and celebrated formulas in mathematics: $\boxed{e^{iy} = \cos y + i\sin y.}$ It tells us that every purely imaginary exponential lies on the unit circle.

This is Euler’s formula. It establishes a profound connection between the exponential function and the trigonometric functions, unifying analysis, algebra, and geometry in a single identity.

As the real parameter $y$ varies, the point $e^{iy}$ traces out the unit circle $|z| = 1$ in the complex plane in the counterclockwise direction (it moves around the circle of radius 1 centered at the origin), starting from $e^{i \cdot 0} = 1$.

The parameter $y$ represents the angle (in radians) measured from the positive real axis: $e^{i \cdot 0} = 1$ (the point $(1, 0)$); $e^{i\pi/2} = i$ (the point $(0, 1)$); $e^{i\pi} = -1$ (the point $(-1, 0)$) — this gives the famous identity $e^{i\pi} + 1 = 0$; $e^{i \cdot 3\pi/2} = -i$ (the point $(0, -1)$); $e^{i \cdot 2\pi} = 1$ (return to the start — periodicity).

The complex exponential function can be viewed as a generalization of the unit circle in the complex plane. So instead of thinking of the unit circle as a geometric object, we can think of it as the image of the imaginary axis under the exponential function:

(i) Real axis $\to$ exponential growth/decay. (ii) Imaginary axis $\to$ circular motion. (iii) Complex plane $\to$ combination of both. The exponential function “wraps” the imaginary axis around the unit circle infinitely many times.

• Polar Form of Complex Numbers. Euler’s formula provides a powerful and compact way to represent complex numbers in polar form. Any complex number $z = x + iy$ with $|z| = r$ and $\arg(z) = \theta$ can be written as $z = x + iy = r\cos(\theta) + ir\sin(\theta) = r(\cos(\theta) + i\sin(\theta))$. Using Euler’s formula, this become $\boxed{z = re^{i\theta}}$ where r = ∣z∣ and $\theta = arg(z)$.

This exponential form is extremely useful for performing operations on complex numbers: (i) Multiplication, $z_1·z_2 = (r_1e^{i\theta_1})(r_2e^{i\theta_2}) = r_1r_2e^{i(\theta_1+\theta_1)}$ (to multiply two complex numbers, we multiply their moduli and add their arguments); (ii) Division, $z_1/z_2 = r_1 e^{i\theta_1} / r_2 e^{i\theta_2} = (r_1/r_2) e^{i(\theta_1 - \theta_2)}$; (iii) Power, $z^n = (re^{i\theta})^n = r^n e^{in\theta}$.

• Complex Trigonometric Functions. Euler’s formula leads to elegant expressions for the trigonometric functions. Euler’s formula states that $e^{i y} = \cos(y) + i·\sin(y)$. If we also write its complex conjugate $e^{-i y} = \cos(y) - i·\sin(y),$ then by adding and subtracting these two expressions we isolate cosine and sine. $\boxed{\cos z = \frac{e^{iz} + e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz} - e^{-iz}}{2i}.}$.

These definitions are consistent with the real-valued functions when z is a real number. Since the exponential function $e^z$ is an entire function, and sums, products, scalar multiples, and compositions of entire functions are also entire, it follows that $\cos(z)$ and $\sin(z)$ are entire functions, too. They are analytic everywhere in the complex plane $\mathbb{C}$.

Unlike their real counterparts, the complex sine and cosine are unbounded. For instance, $\sin(iy) = \frac{e^{-y} - e^y}{2i} = i\frac{e^y - e^{-y}}{2} = i\sinh(y)$, which grows without bound as $y \to \pm\infty$. The familiar inequality $|\sin x| \leq 1$ holds only on the real axis.

• Derivatives of Sine and Cosine. The derivatives of the complex sine and cosine functions can be found by differentiating the exponential definitions using the chain rule and the fact that $\frac{d}{dz}e^z = e^z$.

$\frac{d}{dz}cos(z) = \frac{d}{dz}\frac{eⁱᶻ+e⁻ⁱᶻ}{2} = \frac{ieⁱᶻ-ie⁻ⁱᶻ}{2} = i\frac{eⁱᶻ-e⁻ⁱᶻ}{2} = -\frac{eⁱᶻ-e⁻ⁱᶻ}{2i} = -sin(z)$. Similarly, for the sine function: $\frac{d}{dz}sin(z) = \frac{d}{dz}\frac{eⁱᶻ-e⁻ⁱᶻ}{2i} = \frac{ieⁱᶻ+ie⁻ⁱᶻ}{2i} = \frac{eⁱᶻ+e⁻ⁱᶻ}{2} = cos(z)$. These results are identical to the familiar formulas from real calculus and extend naturally to the complex domain.

• Trigonometric Identities in the Complex Plane. All the standard trigonometric identities that hold for real arguments also hold for complex arguments.

For example, the Pythagorean identity,

$$ \begin{aligned} \cos^2 z + \sin^2 z &=\left(\frac{e^{iz} + e^{-iz}}{2}\right)^{\!2} + \left(\frac{e^{iz} - e^{-iz}}{2i}\right)^{\!2} \\[2pt] &=\frac{e^{2iz} + 2 + e^{-2iz}}{4} + \frac{e^{2iz} - 2 + e^{-2iz}}{-4} \\[2pt] &=\frac{(e^{2iz} + 2 + e^{-2iz}) - (e^{2iz} - 2 + e^{-2iz})}{4}\\[2pt] &=\frac{4}{4} = 1. \end{aligned} $$

Similarly, the angle addition formulas hold:all $z_1, z_2 \in \mathbb{C}$: $\sin(z_1 + z_2) = \sin z_1\cos z_2 + \cos z_1\sin z_2; \cos(z_1 + z_2) = \cos z_1\cos z_2 - \sin z_1\sin z_2; \sin(z_1 - z_2) = \sin z_1\cos z_2 - \cos z_1\sin z_2.$

• Mapping Properties of the Complex Exponential. The complex exponential $w = e^z$ defines a mapping from the $z$-plane to the $w$-plane.

1. Real axis (y = 0). $\{x + 0i : x \in \mathbb{R}\} \xrightarrow{\exp} \{e^x : x \in \mathbb{R}\} = (0, \infty)$ — the positive real axis.
2. Imaginary axis (x = 0). $\{0 + iy : y \in \mathbb{R}\} \xrightarrow{\exp} \{e^{iy} : y \in \mathbb{R}\} = S^1$ - the unit circle.
3. Horizontal line (y = $y_0$), $\{x + iy_0 : x \in \mathbb{R}\} \xrightarrow{\exp} \{e^x e^{iy_0} : x \in \mathbb{R}\}$ - a half‐line (ray) from the origin at angle $y_0$, not including the origin (radius ranges from $0^+$ to $+\infty$).
4. Vertical line (x = $x_0$), $\{x_0 + iy : y \in \mathbb{R}\} \xrightarrow{\exp} \{e^{x_0} e^{iy} : y \in \mathbb{R}\}$ - a full circle of radius $e^{x_0}$ centered at the origin.
   The exponential mapping transforms the orthogonal grid of horizontal and vertical lines in the $z$-plane into an orthogonal grid of rays and concentric circles in the $w$-plane.
5. Horizontal strip $0 \le y < 2\pi$, the image is $\mathbb{C} \setminus \{0\}$, the entire punctured plane. The exponential maps this strip onto $\mathbb{C} \setminus \{0\}$ bijectively: each point in the punctured plane is hit exactly once.
   The mapping $w = e^z$ restricted to any horizontal strip of width $2\pi$ is a bijection onto $\mathbb{C} \setminus \{0\}$.
   This is the geometric manifestation of the $2\pi i$-periodicity: the entire $z$-plane is partitioned into horizontal strips of height $2\pi$, each of which maps onto the same punctured plane.
6. Horizontal trip $\alpha < y < \beta$ ($0 < \beta - \alpha \leq 2\pi$), the image is an open sector between the rays at angles $\alpha$ and $\beta$.
7. Left half‐plane x < 0, the image is $\{w : 0 < |w| < 1\}$ - the open unit disk minus zero.
8. Right half‐plane x > 0, the image is $\{w : |w| > 1\}$, the exterior of the unit disk.
9. Upper half-plane y > 0, the image is the upper half of $\mathbb{C} \setminus \{0\}$.