Justification For Dropping Real Part of s in Transfer Functions, \$F(s)=F(j\omega)\$
If all you are interested in is the bode plot magnitude or phase then sigma is irrelevant and should be made zero so that the resulting formula for the transfer function is defined only for the bode plot plane of the pole zero 3D diagram. Example: -

The resulting bode plot shown below has \$\sigma = 0\$ : -

In other words, it's the \$\sigma = 0\$ plane