Why are functions for the frequency domain expressed in terms of a whole complex number rather than just omega?
Why is this in terms of jω rather than just ω,


In general, we have the complex frequency \$s = \sigma + j\omega\$.

For the Fourier transform, we set \$\sigma = 0\$ such that \$s = j \omega\$.

Thus, \$X(s) \rightarrow X(j\omega)\$

In the case of the Z transform, we have in general \$z = Ae^{j\Omega}\$.

For the Discrete Time Fourier Transform (DTFT), \$A = 1\$.

Thus, \$X(z) \rightarrow X(e^{j\Omega})\$

The notation varies among disciplines and authors.  As long as the context is clear, one can vary the notation.