Let z=x+iy, and w=u+iv. I am looking for a formula to find the arctangent of z, or w=arctan(z). I want the results of u and v to be in terms of trigonometric and hyperbolic functions (and their inverses) and not in terms of logarithms. The values u and v should be functions of x and y.
In other words, a Riemann sphere is covered by complex scalars?
Does this mean that a complex vector resides in a complex space having twice as many real dimensions?
I'm trying to picture a complex vector in my mind. A real vector is usually drawn as an arrow pointing in some direction in space and having a positive real magnitude "length". I'm trying to understand a complex vector the same way.
Is it drawn as an arrow pointing in some direction having a...
If a vector field can be decomposed into a curl field and a gradient field, is there a similar decomposition for scalar fields, say into a divergence field plus some other scalar field?
If all three quarks in a baryon have the same flavor, they all spin the same direction, causing the baryon to have 3/2 spin. In a proton, do both up-quarks spin in the same direction (with the down-quark spinning opposite)? Or can the two up-quarks spin in opposite directions?
Suppose a neutral meson decays into an electron and a positron. Are the two particles entangled as they fly apart? Before any measurement takes place, are the particles in a mixed superposition as to which one is the electron, and which one is the positron? Is there a way to test for such...
Suppose a Fermion having left-handed helicity travels faster than light between points A and B. Some observers see it traveling from A to B. Some observers see it traveling from B to A (backwards in time). Some observers see it traveling instantaneously (infinite speed). Do any any of these...
Why is "generation number" or "family number" conserved among leptons, but not among quarks? Why does a positive kaon decay into an antimuon and a muon neutrino? Why doesn't it decay into an antimuon and an electron neutrino? The anti-strange (generation 2) flavor then becomes anti-muon flavor...
If the top quark is heavier than the Higgs boson, then why was it discovered first? Does the top quark get all of its mass from the Higgs? Or does some of its mass come from somewhere else, perhaps a heavier Higgs boson, not yet discovered?
I've read that if right-handed neutrinos exist, they should be heavier than left-handed neutrinos (why?). I figured if an observer moves faster than a left-handed neutrino, he'd see it as right-handed. Also, would right-handed neutrinos interact with Z-bosons? How would such interactions be...
I wasn't sure whether to post this in the classical forum or the quantum forum. In "SU(3) Maxwell equations and the Classical Chromodynamics" by Sanchez Monroy, he gives four field equations for CCD similar to Maxwell's four field equations for EM fields. However, unlike Maxwell's equations...