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?

Yes, all complex expressions that produce a complex number ($\arctan(i)$ doesn't) can be written in rectangular form. You can generally use a Taylor Series or convert your expression to another form to find the complex number in rectangular form. Here is a page that deals with a lot of these cases. Most of these results use $e^{ix}=\cos(x)+i\sin(x)$ to convert functions that don't seem to work with complex numbers into ones that do.
