Aðalgildi

Aðalgildi is an Icelandic term that translates to "main value" or "principal value." It is most commonly encountered in the context of mathematics, particularly in calculus and complex analysis. When dealing with multi-valued functions, such as the complex logarithm or inverse trigonometric functions, a convention is established to select a single, specific value to be considered the "principal value." This selection is often based on a defined range or branch cut, ensuring that the function behaves in a consistent and predictable manner.

For example, the complex logarithm of a non-zero complex number $z = r e^{i\theta}$ can be expressed as $\ln(z) = \ln(r) + i(\theta + 2\pi k)$, where $k$ is any integer. The principal value of the complex logarithm, denoted as $\text{Ln}(z)$, is obtained by restricting the argument $\theta$ to a specific interval, typically $(-\pi, \pi]$. This choice defines the principal branch of the logarithm. Similarly, the principal value of the square root function for a complex number $z$ is often chosen such that its argument lies in the interval $[0, \pi)$. The concept of the principal value is crucial for defining inverse functions that are single-valued and for avoiding ambiguities in mathematical operations.