Complex logarithm

In mathematics, a complex logarithm is a way to extend the idea of the natural logarithm to work with complex numbers. Regular logarithms only work with positive real numbers, but complex logarithms can be used for any nonzero complex number.

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, 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. Mathematicians use special versions of the logarithm called "branches" or define a "principal value" to pick one specific logarithm for each complex number. These ideas connect to deeper topics like analytic continuation, 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 single result. For example, adding a certain special number (like 2πi) to a complex number doesn’t change the result. This means many different inputs give the same output, so we can’t just 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 keeps the angles between lines the same. The complex logarithm is smooth and its change rate (1/z) is never zero when (z) is not zero. This makes it a conformal map. It keeps the shapes and angles of objects in the complex plane.

The main version of the logarithm, where the imaginary part of the angle (\theta) is between (-\pi) and (\pi), shows how the logarithm works in different parts of the complex plane. It helps us see how the logarithm changes angles and shapes in special ways.

Applications

The complex logarithm helps us work with complex numbers when we want to raise them to powers. If we need 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 can give us many possible results, depending on which logarithm we pick.

Complex logarithms are also important in trigonometry. Many trigonometric functions can be written using exponentials, and their inverse functions can be shown using complex logarithms. In areas like electrical engineering, complex logarithms are used 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 is done by using the natural logarithm and changing it a little. The exact value can change depending on how we define the logarithm for the base number.

When working 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 in the complex plane.