I am trying to determine the transfer function $$H(s)=v_{OUT}(s)/v_{in}(s)$$, zeros and poles of the transfer function of the above circuit.
From the obtained form of the transfer function H(s) as the voltage gain $$H(s)=K_{U}(s)=v_{OUT}(s)/v_{in}(s)$$, I am not able to determine its zeros and poles.
$$Z_{1}=R_{2}\\Z_{2}=\frac{1}{sC_{2}}\parallel R_{3}=\frac{\frac{1}{sC_{2}}\cdot R_{3}}{\frac{1}{sC_{2}}+ R_{3}}=\frac{\frac{R_{3}}{sC_{2}}}{\frac{1+sC_{2}R_{3}}{sC_{2}}}=\frac{R_{3}}{sC_{2}}\cdot \frac{sC_{2}}{1+sC_{2}R_{3}}=\frac{R_{3}}{1+sC_{2}R_{3}}\\E_{TH}=E\frac{Z_{2}}{Z_{1}+Z_{2}}=E\frac{\frac{R_{3}}{1+sC_{2}R_{3}}}{R_{2}+\frac{R_{3}}{1+sC_{2}R_{3}}}=E\frac{\frac{R_{3}}{1+sC_{2}R_{3}}}{\frac{R_{2}(1+sC_{2}R_{3})+R_{3}}{1+sC_{2}R_{3}}}=E\frac{R_{3}}{1+sC_{2}R_{3}} \frac{1+sC_{2}R_{3}}{R_{2}+sC_{2}R_{3}R_{2}+R_{3}}=E\frac{R_{3}}{R_{2}+R_{3}+sC_{2}R_{2}R_{3}}\\\\Y_{R_{2}}=\frac{1}{R_{2}}\\Y_{R_{3}}=\frac{1}{R_{3}}\\Y_{C_{2}}=sC_{2}\\Y_{TH}=Y_{R_{2}}+Y_{R_{3}}+Y_{C_{2}}=\frac{1}{R_{2}}+\frac{1}{R_{3}}+sC_{2}=\frac{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}{R_{2}R_{3}}\\Z_{TH}=\frac{1}{Y_{TH}}=\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}\\\\Z_{C_{1}}=\frac{1}{sC_{1}}\\v_{in}(s)=\frac{1}{sC_{1}}i(s)+R_{1}i(s)+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}i(s)+E_{TH}\\v_{OUT}(s)=R_{1}i(s)+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}i(s)+E_{TH}\\K_{U}(s)=\frac{v_{OUT}(s)}{v_{in}(s)}=\frac{R_{1}i(s)+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}i(s)+E_{TH}}{\frac{1}{sC_{1}}i(s)+R_{1}i(s)+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}i(s)+E_{TH}} =\frac{i(s)\cdot (R_{1}+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}+\frac{E_{TH}}{i(s)})}{i(s)\cdot (\frac{1}{sC_{1}}+R_{1}+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}+\frac{E_{TH}}{i(s)})} =\frac{R_{1}+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}+\frac{E_{TH}}{i(s)}}{\frac{1}{sC_{1}}+R_{1}+\frac{R_{2}R_{3}}{R_{3}+R_{2}+sC_{2}R_{2}R_{3}}+\frac{E_{TH}}{i(s)}}$$