Argument (complex analysis) in the context of "Imaginary number"

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).

In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point.

Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link that was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.

In mathematics, the set of positive real numbers, ${\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},}$ is the subset of those real numbers that are greater than zero. The non-negative real numbers, ${\displaystyle \mathbb {R} _{\geq 0}=\left\{x\in \mathbb {R} \mid x\geq 0\right\},}$ also include zero. Although the symbols ${\displaystyle \mathbb {R} _{+}}$ and ${\displaystyle \mathbb {R} ^{+}}$ are ambiguously used for either of these, the notation ${\displaystyle \mathbb {R} _{+}}$ or ${\displaystyle \mathbb {R} ^{+}}$ for ${\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}}$ and ${\displaystyle \mathbb {R} _{+}^{*}}$ or ${\displaystyle \mathbb {R} _{*}^{+}}$ for ${\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}}$ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.

In a complex plane, ${\displaystyle \mathbb {R} _{>0}}$ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers ${\displaystyle z=|z|\mathrm {e} ^{\mathrm {i} \varphi },}$ with argument ${\displaystyle \varphi =0.}$