(1)

\(C_{1}\)は正の実軸の上側を無限から0までマイナス方向に進む経路
\(C_{2}\)は原点の回りを反時計回りで1周する経路
\(C_{3}\)は正の実軸の下側を0から無限までプラス方向に進む経路
とする。
\begin{align*} \zeta\left(s,\alpha\right) & =\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{t^{s-1}e^{-\alpha t}}{1-e^{-t}}\right)dt\\ & =\frac{\sin\left(\pi s\right)\Gamma\left(1-s\right)}{\pi}\int_{0}^{\infty}\left(\frac{t^{s-1}e^{-\alpha t}}{1-e^{-t}}\right)dt\\ & =\frac{\left(e^{i\pi s}-e^{-i\pi s}\right)\Gamma\left(1-s\right)}{2\pi i}\int_{0}^{\infty}\left(\frac{t^{s-1}e^{-\alpha t}}{1-e^{-t}}\right)dt\\ & =-\frac{\left(e^{i\pi\left(s-1\right)}-e^{-i\pi\left(s-1\right)}\right)\Gamma\left(1-s\right)}{2\pi i}\int_{0}^{\infty}\left(\frac{t^{s-1}e^{-\alpha t}}{1-e^{-t}}\right)dt\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\left(\int_{0}^{\infty}\frac{\left(e^{i\pi}t\right)^{s-1}e^{-\alpha t}}{1-e^{-t}}dt+\int_{\infty}^{0}\frac{\left(e^{-i\pi}t\right)^{s-1}e^{-\alpha t}}{1-e^{-t}}dt\right)\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\left(\int_{0}^{\infty}\frac{\left(e^{2i\pi}e^{-i\pi}t\right)^{s-1}e^{-\alpha t}}{1-e^{-t}}dt+\int_{\infty}^{0}\frac{\left(e^{-i\pi}t\right)^{s-1}e^{-\alpha t}}{1-e^{-t}}dt\right)\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\left(\int_{C_{3}}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz+\int_{C_{1}}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz\right)\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\left(\int_{C_{3}}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz+\int_{C_{1}}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz+\int_{C_{2}}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz\right)\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{C}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz \end{align*}

(2)

\begin{align*} \zeta\left(s\right) & =\zeta\left(s,1\right)\\ & =-\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{C}\frac{\left(-z\right)^{s-1}e^{-z}}{1-e^{-z}}dz \end{align*}