The phase of a complex number represented as a = C * exp(i φ) is defined as φ when C > 0. However, the phase is not uniquely defined due to the periodic nature of the exponential function, leading to the equivalence C * exp(i φ) = C * exp(i(φ + 2πn)) for any integer n. To resolve this ambiguity, the phase can be defined within a coset φ + 2πℤ or constrained to intervals such as [0, 2π) or [-π, π). For C < 0, the phase must incorporate the sign of C, resulting in a phase of φ + π. When C = 0, the phase is undefined.
PREREQUISITESMathematicians, physicists, engineers, and students studying complex analysis or signal processing who need a clear understanding of the phase of complex numbers.
If ##C > 0## then this is almost correct. However, the phase is not well-defined under this definition, because ##C\exp(i\phi) = C\exp(i(\phi+2\pi n))## for any integer ##n##. You can get around this by defining the phase to be the coset ##\phi + 2\pi \mathbb{Z}## or by constraining it to be in the interval ##[0,2\pi)## or ##[-\pi, \pi)## or some other half-open interval of length ##2\pi##.pamparana said:I just wanted to check something. If I have a complex number of the form
a = C * \exp(i \phi)
where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?