Elementary Analytic Functions: Exponential and Logarithm
students, this lesson introduces two of the most important functions in complex analysis: the complex exponential function and the complex logarithm. These functions connect algebra, geometry, and calculus in a powerful way πβ¨. By the end of this lesson, you should be able to explain what these functions are, how they behave in the complex plane, and why branch issues appear when we try to define logarithms for complex numbers.
What makes the complex exponential special?
In real calculus, the exponential function is familiar as $e^x$, where $e$ is the base of the natural logarithm. In complex analysis, we extend this idea to complex inputs by defining the complex exponential function as
$$e^z = e^{x+iy} = e^x(\cos y + i\sin y),$$
where $z = x + iy$ with real numbers $x$ and $y$. This formula is not just a definition for convenience. It comes from the power series
$$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!},$$
which converges for every complex number $z$. That means $e^z$ is defined everywhere in the complex plane and is an entire function, meaning it is analytic at every point.
This formula shows two very important facts:
- The real part $x$ controls the size, because $e^x$ is always positive.
- The imaginary part $y$ controls rotation, because
cos y$ and $
sin y describe a point on the unit circle.
So if $z = x + iy$, then $e^z$ stretches a point by a factor of $e^x$ and rotates it by angle $y$ radians. This makes the exponential function both algebraic and geometric at the same time π―.
For example, if $z = i\pi$, then
$$e^{i\pi} = \cos \pi + i\sin \pi = -1.$$
This is part of the famous identity
$$e^{i\pi} + 1 = 0,$$
which links $e$, $i$, $\pi$, $1$, and $0$ in one compact formula.
Key properties of $e^z$
students, the complex exponential satisfies many familiar rules from real exponentials, and these remain true in the complex setting:
$$e^{z+w} = e^z e^w,$$
$$\frac{d}{dz}e^z = e^z.$$
The first identity is very useful because it shows exponential growth and rotation combine nicely. The second identity is one reason $e^z$ plays a central role in differential equations and analytic function theory.
However, there is one major difference from the real case: the complex exponential is periodic in the imaginary direction. Since
$$e^{z+2\pi i} = e^z,$$
many different complex numbers give the same exponential value. For instance,
$$e^{i\theta} = e^{i(\theta + 2\pi k)}$$
for any integer $k$. This repeating behavior is the source of the branch issues that appear later when we define logarithms.
A useful geometric picture is this: the map $z \mapsto e^z$ sends vertical lines to circles and horizontal lines to rays. For example, if $z = x + iy$ and $x$ is fixed, then $|e^z| = e^x$ is constant, so the image is a circle centered at the origin. If $y$ is fixed, then the argument stays fixed and the image is a ray.
The complex logarithm: inverse idea, new complications
In real algebra, logarithms undo exponentials. In complex analysis, the same idea works only with care. We want a function $w = \log z$ such that
$$e^w = z.$$
Write a nonzero complex number in polar form:
$$z = re^{i\theta},$$
where $r > 0$ and $\theta$ is an argument of $z$. Then a possible logarithm is
$$\log z = \ln r + i\theta.$$
But here is the crucial issue: the angle $\theta$ is not unique. Since
$$re^{i\theta} = re^{i(\theta + 2\pi k)},$$
for any integer $k$, we actually get infinitely many logarithms:
$$\log z = \ln r + i(\theta + 2\pi k), \quad k \in \mathbb{Z}.$$
So unlike the real logarithm, the complex logarithm is multivalued. This is the main branch issue in this topic.
For example, for $z = -1$, we can write
$$-1 = e^{i(\pi + 2\pi k)},$$
so
$$\log(-1) = i(\pi + 2\pi k), \quad k \in \mathbb{Z}.$$
There is no single complex number that is the one and only logarithm of $-1$ unless we choose a rule to select just one value.
Principal branch and branch cuts
To make the logarithm into a single-valued function, we choose one preferred angle. A common choice is the principal argument, usually written $\operatorname{αα α} z$ or more commonly $\operatorname{Arg} z$, where
$$-\pi < \operatorname{Arg} z \le \pi.$$
Then the principal branch of the logarithm is defined by
$$\Log z = \ln|z| + i\operatorname{Arg} z, \quad z \ne 0.$$
This gives one specific value for each nonzero complex number, but it is not continuous on all of $\mathbb{C}\setminus\{0\}$. To keep the argument from jumping suddenly, we remove a curve called a branch cut, often the nonpositive real axis. On the remaining region, the principal logarithm is analytic.
Why do we need a branch cut? Imagine walking around the origin in a loop. The angle changes continuously, but after one full turn, it increases by $2\pi$. That means the logarithm cannot return to exactly the same value if we insist on a continuously changing argument everywhere around $0$. The origin is the source of the obstruction.
This is a major idea in complex analysis: some functions naturally live on domains with holes, and to define them consistently we often need to choose branches. The logarithm is one of the clearest examples of this phenomenon π.
Examples that show how the logarithm works
Letβs look at a few examples.
Example 1: logarithm of a positive real number
If $z = 5$, then $|z| = 5$ and $\operatorname{Arg} z = 0$. So the principal logarithm is
$$\Log 5 = \ln 5.$$
This matches the ordinary real logarithm.
Example 2: logarithm of a negative real number
If $z = -3$, then $|z| = 3$ and the principal argument is $\pi$. So
$$\Log(-3) = \ln 3 + i\pi.$$
Other values are also possible:
$$\log(-3) = \ln 3 + i(\pi + 2\pi k), \quad k \in \mathbb{Z}.$$
Example 3: logarithm of a pure imaginary number
If $z = 2i$, then $|z| = 2$ and a principal argument is $\pi/2$. Therefore
$$\Log(2i) = \ln 2 + i\frac{\pi}{2}.$$
These examples show the two parts of the logarithm clearly: the real part is $\ln|z|$, and the imaginary part comes from the angle.
Connection to analytic functions and differentiation
students, the exponential function is entire, but the logarithm is not entire because it cannot be defined as a single-valued analytic function on all of $\mathbb{C}\setminus\{0\}$. On a suitable domain that avoids a branch cut, however, a branch of the logarithm is analytic.
For the principal branch, if $z$ is in the chosen domain, then
$$\frac{d}{dz}\Log z = \frac{1}{z}.$$
This is the same derivative as in real calculus, but it only works where the branch is analytic and $z \ne 0$.
The exponential and logarithm are inverse functions in a limited sense. On the principal branch domain,
$$\Log(e^z)$$
need not equal $z$ for every complex number $z$, because of the $2\pi i$ periodicity. But if $z$ is chosen with imaginary part in the principal interval, then the identity behaves as expected. Similarly,
$$e^{\Log z} = z$$
for every $z$ in the domain of the chosen branch.
This mismatch is another way to see why branch issues matter. In complex analysis, inverse functions are often only local inverses rather than global inverses.
Why this matters in the bigger topic of elementary analytic functions
Exponential and logarithm are part of a larger family of elementary analytic functions, along with trigonometric and hyperbolic functions. In fact, many of those functions are built from exponentials. For example,
$$\sin z = \frac{e^{iz} - e^{-iz}}{2i},$$
and
$$\cos z = \frac{e^{iz} + e^{-iz}}{2}.$$
Because of these formulas, understanding $e^z$ helps you understand the complex trigonometric and hyperbolic functions too.
The complex logarithm is also important in solving equations such as
$$e^z = w,$$
because taking a logarithm gives the family of solutions
$$z = \Log w + 2\pi i k, \quad k \in \mathbb{Z},$$
when the appropriate branches are considered. This kind of reasoning appears in equation solving, contour analysis, and the study of multi-valued functions.
Conclusion
The complex exponential function $e^z$ is an entire function defined by a power series and expressed geometrically by
$$e^{x+iy} = e^x(\cos y + i\sin y).$$
The complex logarithm tries to reverse the exponential map, but because angles in the complex plane repeat every $2\pi$, it is inherently multivalued:
$$\log z = \ln|z| + i(\operatorname{Arg} z + 2\pi k).$$
To make it single-valued, we choose a branch such as the principal logarithm $\Log z$. Branch cuts are needed so the chosen angle stays continuous. Together, the exponential and logarithm form a foundation for many other elementary analytic functions in complex analysis. Mastering them gives you a strong base for studying trigonometric functions, hyperbolic functions, and more advanced topics π.
Study Notes
- The complex exponential is defined by $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ and is entire.
- If $z = x + iy$, then $e^z = e^x(\cos y + i\sin y)$.
- The exponential is periodic: $e^{z+2\pi i} = e^z$.
- The complex logarithm is an inverse idea, but it is multivalued because arguments differ by $2\pi$.
- For $z = re^{i\theta}$, the general logarithm is $\log z = \ln r + i(\theta + 2\pi k)$, where $k \in \mathbb{Z}$.
- The principal logarithm is $\Log z = \ln|z| + i\operatorname{Arg} z$, with $-\pi < \operatorname{Arg} z \le \pi$.
- A branch cut is used to make a chosen logarithm single-valued and analytic on a restricted domain.
- The derivative of a chosen analytic branch is $\frac{d}{dz}\Log z = \frac{1}{z}$.
- Exponential and logarithm are building blocks for complex trigonometric and hyperbolic functions.
- Branch issues are a central feature of complex analysis because many complex functions are naturally multivalued.