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?
A complex number is a number that contains a real part and an imaginary part. It is usually written in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.
A complex number is typically represented on a graph known as the complex plane, where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis.
The complex number Z = u^v is defined as the result of raising the complex number u to the power of the complex number v. This can be written as Z = u^v = e^(v*ln(u)), where e is the base of the natural logarithm and ln(u) is the natural logarithm of u.
The modulus of a complex number is the distance of the complex number from the origin on the complex plane. It is calculated by taking the square root of the sum of the squares of the real and imaginary parts, and is denoted by |Z|.
The conjugate of a complex number is another complex number that has the same real part but the opposite sign for the imaginary part. It is denoted by Z* and is calculated by changing the sign of the imaginary part of the complex number.