In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, domain coloring allows for a function from the complex plane to itself, whose graph would normally require four spatial dimensions, to be easily represented and understood. This provides insight to the fluidity of complex functions and shows natural geometric extensions of real functions.
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Domain coloring in the context of Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for Re(s) > 1, and its analytic continuation elsewhere.
The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Domain coloring in the context of Atomic orbital
In quantum mechanics, an atomic orbital (/ˈɔːrbɪtəl/ ) is a function describing the location and wave-like behavior of an electron in an atom. This function describes an electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.
Each orbital in an atom is characterized by a set of values of three quantum numbers n, ℓ, and mℓ, which respectively correspond to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of mℓ and −mℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy, x − y) which describe their angular structure.