Rc filter transfer function
No way to be 100% sure of the correctness, but some validations can be done:

1) calculation of units/dimmension

You add 1 (adimensional) and \$j\omega R_1C\$ thus \$\omega R_1C\$ must be adimensional. It is, pass check 1.

The global function has unit \$ \Omega^{-1} \$, this is not valid for a transfer function of type \$ V_{out}/V_{in} \$ (that must be dimensionless) but could be valid for \$ I_{out}/V_{in} \$. Suspicious. What kind of transfer function are you calculating? Should it be \$ \frac{R_1}{R_1+R_2} \$ ?

2) Do some limit checks.

Your transfer function is \$ 1/R_2 \$, independent of \$ R_1, \$ when w=0. Is it in the circuit ? Suspicious.

Your transfer function is 0 when \$ R_2 = \infty \$. Could you check that in the original circuit?

...

* Addendum: second case *

The second case that is included in the question is dimensionless and internally coherent, thus, it is a valid transfer function for a system Vout/Vin or similar. That doesn't means it is correct, but it has the correct dimensions.