Complex logarithm

In mathematics, a complex logarithm is a way to extend the idea of the natural logarithm to work with complex numbers. Unlike regular logarithms that only work with positive real numbers, complex logarithms can be used for any nonzero complex number. This makes them very useful in many areas of math and physics where complex numbers are important.

For any nonzero complex number z, a complex logarithm is another complex number w such that when you raise the mathematical constant e to the power of w, you get back to z. This is written as e^w = z. If z is written in a special form called polar form, which uses a distance from zero (called r) and an angle (called θ), then one possible logarithm of z is given by a combination of the natural logarithm of r and the angle θ.

However, unlike regular logarithms, complex logarithms have many possible values for the same z. These values are spaced evenly along a vertical line in the complex plane. This means there isn't just one answer, but many answers that differ by multiples of 2πi. To handle this, mathematicians use special versions of the logarithm called "branches" or define a "principal value" which picks one specific logarithm for each complex number, except for numbers along the negative real axis. These ideas connect to deeper topics like analytic continuation and Riemann surfaces, showing how complex logarithms fit into the larger picture of complex analysis.

Problems with inverting the complex exponential function

The complex exponential function is special, but it has a problem: it doesn’t always give a unique result. For example, adding a certain special number (like 2πi) to a complex number doesn’t change the result of the exponential function. This means many different inputs give the same output, so we can’t simply find an inverse.

To solve this, mathematicians use two main ideas. One is to limit the inputs to a smaller range, which helps pick just one answer for each input — these are called branches. The other idea is to use a more advanced concept called a Riemann surface, which collects all possible answers together. Each method has its own uses, but both help us work with logarithms in the complex plane.

Main article: Branches

Principal value

The principal value of a complex logarithm is a special way to find the logarithm of a nonzero complex number. For any such number ( z ), the principal value, written as ( \operatorname{Log} z ), is the logarithm whose imaginary part is between ( -\pi ) and ( \pi ). This means it fits within a specific range, making it easier to work with.

To calculate the principal value, we use the polar form of the complex number. This form expresses the number as ( r e^{i\theta} ), where ( r ) is the distance from the origin (called the absolute value), and ( \theta ) is the angle (called the argument). The principal value of the logarithm is then given by ( \ln r + i\theta ), where ( \theta ) is chosen to be within the range ( (-\pi, \pi] ). This helps ensure consistency and simplicity in calculations.

The complex logarithm as a conformal map

A conformal map preserves angles between curves. Because a branch of the complex logarithm is holomorphic (smooth) and its derivative (1/z) is never zero for nonzero (z), it acts as a conformal map. This means it keeps the shapes and angles of figures in the complex plane.

The principal branch of the logarithm, where the imaginary part of the angle (\theta) lies between (-\pi) and (\pi), maps the complex plane minus the non-positive real axis to a horizontal strip. This mapping helps visualize how the logarithm behaves across different parts of the complex plane, showing its unique properties in transforming angles and shapes.

Applications

The complex logarithm helps us understand how to raise complex numbers to powers. When we want to calculate something like a^b where a and b are complex numbers and a is not zero, we can use the complex logarithm to find the answer. This gives us many possible results, depending on which logarithm we choose.

Complex logarithms are also useful in trigonometry. Many trigonometric functions can be written using exponentials, and their inverse functions can be expressed using complex logarithms. In fields like electrical engineering, complex logarithms appear in formulas for things like the propagation constant.

Generalizations

Just like with regular numbers, we can find logarithms for complex numbers using different bases. This works by using the natural logarithm and adjusting it, though the exact value can change depending on how we define the logarithm for the base number.

When dealing with special mathematical functions called holomorphic functions, we can also define a version of the logarithm. This helps us understand how these functions behave and change across different points in the complex plane.