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?

Assume $z=re^{i\theta}=x+iy$. Then
$$\ln(z)=\ln(r)+\ln(e^{i\theta})=\ln(r)+i\theta=\ln(\sqrt{x^2+y^2})+i\arctan(\frac{y}{x})$$
Also regarding $\arctan(z)$, it is not difficult to show that $$\arctan(z)=\frac{1}{2i}\ln\left(\frac{i-z}{i+z}\right)$$
