第3問

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