Electrical – Multi-feedback active bandpass filter – transfer function derivation

Assuming an ideal op-amp, you have \$v_1=v_2\$ (as pointed out in a comment by LvW). Also note that your second equation is wrong because there is a current into or out of the output of the op-amp, so you can't just add up the impedances of \$R_2\$ and \$C_2\$.

Introducing another unknown voltage \$v_x\$ at the output of the op-amp, you can write down three equations:

$$(v_i-v_1)sC_1-v_1/R_3=0\\ (v_i-v_1)/R_1+(v_x-v_1)/R_2=0\\ (v_i-v_o)/R_4-(v_o-v_x)sC_2=0$$

where \$v_i\$ and \$v_o\$ are the input and output voltages, respectively, and \$v_1\$ is the voltage at both inputs of the op-amp. These equations can be solved for the transfer function \$H(s)\$, i.e., for the ratio \$v_o/v_i\$. With \$C=C_1=C_2\$ and \$R=R_3=R_4\$ and \$R_1=R_2\$ you get

$$H(s)=R\frac{s^2+\frac{1}{R^2C^2}}{s^2+s\frac{2}{RC}+\frac{1}{R^2C^2}}\tag{1}$$

From the numerator of \$(1)\$ you immediately get for the notch frequency

$$\omega_0=\frac{1}{RC}\tag{2}$$

Note that as long as you choose \$R_1=R_2\$, the actual value of these resistors doesn't show up in the transfer function.

A contribution: Using the Thevenin's Theorem it is possible to transform the circuit to a simpler one:

Using the superposition principle, the output \$V_{out}\$ is:

$$ V_{out}=\left [ 1 + \frac{Z_f(R_1+R_0)}{R_0R_1+R_0R_3+R_1R_3}\right ]V_{ref} - \frac{Z_fR_0}{R_0R_1+R_0R_3+R_1R_3}V_{in}$$

You can't get a transfer function in closed form relating \$V_{out}\$ and \$V_{in}\$. This situation is similar to what happens in control systems, when the plant has two inputs: Reference and disturbance. In that case, the principle of superposition can be used, resulting in two separate transfer functions: one relating the output to the reference input and the other relating the output to the disturbance input.

However, assuming that \$V_{ref}\$ is constant, a transfer function related to input with the output can be obtained, considering the ratio between the variations of these quantities (linear relationship):

$$ \frac{\Delta V_{out}}{\Delta V_{in}} = G_1$$

where:

$$ G_1 = - \frac{Z_fR_0}{R_0R_1+R_0R_3+R_1R_3}$$