The discussion revolves around defining a complex number \( z \) in terms of two variables \( u \) and \( v \), specifically exploring how to express \( u \) and \( v \) in terms of the polar representation of \( z \) given by \( z = r \exp(i \theta) \). The conversation includes attempts to relate these variables to Cartesian coordinates as well.
Participants do not reach a consensus on the definitions of \( u \) and \( v \) or how to express them in terms of \( r \) and \( \theta \). Multiple competing views and interpretations remain present throughout the discussion.
Some participants highlight the ambiguity in the definitions of \( u \) and \( v \), and there are unresolved questions about the relationship between these variables and the standard forms of complex numbers.
Have you looked at this article?Bruno Tolentino said:If I define the complex number z = r exp(i θ) how z = uv, so, how to express u and v in terms of r and θ?
u(r, θ) = ?
v(r, θ) = ?
And the inverse too:
r(u, v) = ?
θ(u, v) = ?
I'm not sure what you are looking for, but [itex]u=re^{i\theta},\ v=1[/itex] works.Bruno Tolentino said:If I define the complex number z = r exp(i θ) how z = uv, so, how to express u and v in terms of r and θ?
u(r, θ) = ?
v(r, θ) = ?
And the inverse too:
r(u, v) = ?
θ(u, v) = ?
Well, let's try. You know that [itex]z=x+iy[/itex] can also be expressed as [itex]z=re^{i\phi}[/itex], where [itex]r=\sqrt{x^{2}+y^{2}}[/itex] and [itex]\phi = \arcsin(\frac{y}{r})[/itex]. Therefore, we obviously have [itex]z=e^{\ln r}\cdot e^{i\phi}= e^{\ln r + i\phi}[/itex]. ...Bruno Tolentino said:So, is possible convert the expression z = x + i y in z = uv ? Is possible express u and v in terms of x and y?