What is the correct notation for flipping $a$ and $b$ values in a complex?

Question

I'm currently doing some experiments on fractals and in one of my equation I need to flip the real and imaginary components of a complex number, such as :

$$ z = a + bi $$

Becomes :

$$ z = b + ai $$

What would be the correct mathematical term or notation for this ? flip ?

Answer 1

As per the comments, if $$z = a + bi$$ then $$i\cdot\bar z = b+ai$$ where $\bar z$ is the conjugate of $z$.

Answer 2

I don't know such notation, but note that in general, $$(a+bi)\times \left(\frac{2ab}{a^2+b^2}+\frac{a^2-b^2}{a^2+b^2}i\right)=b+ai$$ where $(a,b)\not=(0,0).$

Answer 3

If you represent the complex no. as: $$z=a+bi=\sqrt{a^2+b^2}e^{i\arctan(b/a)}$$

Then the flipped number can be written as:

$$z'=b+ai=\sqrt{a^2+b^2}e^{i\cdot(\pi/2-arctan(b/a))}$$