Can all complex expressions be simplified to the form $a+jb$? Are there any complex expressions that cannot be simplified to the form $a+jb$, where a and b are real numbers?
For example, $$\frac{1}{j}=0+j(-1),\hspace{0.5cm}e^j=\cos(1)+j\sin(1),\hspace{0.5cm}\sin(j)=0+j\frac{e^2-1}{2e}$$
From what I understand, all complex numbers must exist somewhere on the complex plane where a and b are the coordinates. But some expressions don't have any obvious way to be simplified:
$$\ln(1+j)=???,\hspace{0.5cm}\arctan(j)=???$$
If every expression can be simplified, are there any good references or list of identities?

To get
$\ln(z)$
for any complex $z$
(where $z \ne 0$),
write
$z = |z|y$.
Then $|y| = 1$,
so there is a real $t$
such that
$y = e^{it}
=\cos(t)+i\sin(t)
$,
so
$z 
= |z|y
= |z|(\cos(t)+i\sin(t))
= |z|\cos(t)+i|z|\sin(t)
$.
Since
$|1+j| = \sqrt{2}$,
$(1+j)
=\sqrt{2}\frac{1+j}{2}
=\sqrt{2}e^{i\pi/4}
$,
so
$\ln(1+j)
=\ln(\sqrt{2}e^{i\pi/4})
=\dfrac{\ln(2)}{2}+i\pi/4
$
(you can throw in
$+2\pi  i n$
if you want).
