Choose a vector on the complex plane that is represented in the conventional form of $Z = (a + i \space b )$. Have the "length" associated with this complex number be 1. Set the rotation of the complex number as $\frac{\pi}{4}$ radians above the real axis; to wit, have:
$$ Z_0 = cos(\frac{\pi}{4}) + i\space sin(\frac{\pi}{4}) $$
Now change the angle from ($\frac{\pi}{4}$) to ($ i \space \frac{\pi}{4}$). The expression then becomes:
$$ Z_1 = cos(\frac{i\space\pi}{4}) + i\space sin(\frac{i\space\pi}{4}) $$
In general, what transformation will you see happen to this vector on the complex plane? How does the length, angle, and location of the vector change? Also, can this transformation from $Z_0$ to $Z_1$ be represented as two functions $f$ and $g$ on the real and imaginary parts of $Z_0$? Concretely:
$$ Z_1 = f \circ Real(Z_0) + i \times(g \circ Imaginary(Z_0)) $$
I computed this value and had $Z_0$ equal to ($0.71 + i \space 0.71$) and $Z_1$ equal to ($0.46 + i \space 0$). I am stuck on visualizing the general transformation of what any generalized vector at angle ($x$) is subjected to when the angle goes to ($i \space x$)
Many thanks in advance.