第3問

以下の式を求めてください．ただし$\alpha \in \mathbb{C}$，$\beta \in \mathbb{C}\mathrm{\setminus }\left\{0\right\}$，$f:\mathbb{C}\to \mathbb{C}$は整関数，$\gamma :[0,1]\to \mathbb{C}$は連続関数とし，
${\displaystyle \mathrm{\forall }x\in \mathbb{R},f(x+i\mathfrak{I}\alpha )\in \mathbb{R}}$
${\displaystyle \mathrm{\forall }y\in \mathbb{R},f(\mathfrak{R}\alpha +iy)\in \mathbb{R}}$
${\displaystyle \gamma \left(0\right)=\alpha -\beta }$
${\displaystyle \gamma \left(1\right)=\alpha +\beta }$
${\displaystyle \mathrm{\forall }t\in ]0,1[,\mathfrak{I}\frac{\gamma \left(t\right)-\alpha }{\beta }<0}$
を満たすとします．
${\displaystyle {\int }_{\gamma }\frac{f\left(z\right)}{z-\alpha }dz}$