Dick said:Using a '*' for complex conjugation is more usual than '~'.
"Conjugate" refers to the complex conjugate of a complex number, which involves changing the sign of the imaginary part. For example, the complex conjugate of 3+4i is 3-4i.
A complex number is convergent if its real and imaginary parts both approach a finite limit as the independent variable approaches a certain value. In other words, the real and imaginary parts of the complex number must become closer and closer to a constant value as the independent variable gets closer to a specific value.
Yes, the statement "If a complex converges, then it's conjugate converges" applies to all complex numbers, as long as the complex number in question is convergent.
No, if a complex number is convergent, then its conjugate must also be convergent. This is because the real and imaginary parts of a complex number are dependent on each other, and if one part is converging, the other part must also be converging in order for the entire complex number to converge.
This statement is important in complex analysis, as it helps to determine whether a function is analytic (differentiable) at a point. If a complex number and its conjugate both converge, then the function is analytic at that point. This statement also has applications in the study of complex series and their convergence.