広義積分(improper integrals)

演習問題

1.: 次の広義積分を求めよう．
(a): $\displaystyle{\iint_{\Omega}\frac{dxdy}{(y^2 - x^2)^{1/2}}, \Omega = \{(x,y) : 0 \leq x < y \leq 1\}}$
(b): $\displaystyle{\iint_{\Omega}e^{-(x+y)}dxdy, \Omega = \{(x,y) : x \geq 0, y \geq 0\}}$
(c): $\displaystyle{\iint_{\Omega}\tan^{-1}(\frac{y}{x})dxdy, \Omega = \{(x,y) : x,y \geq 0, x^2 + y^2 \leq 1 \}}$
(d): $\displaystyle{\iint_{\Omega}\frac{1}{x^2 y^2}dxdy, \Omega = \{(x,y) : x \geq 1, y \geq 1 \}}$
(e): $\displaystyle{\iint_{\Omega}\frac{dxdy}{\sqrt{x - y^2}}, \Omega = \{(x,y) : 0 \leq x \leq 1, y^2 \leq x \}}$
(f): $\displaystyle{\iint_{\Omega}\frac{dxdy}{1 + (x^2 + y^2)^2}, \Omega = \{(x,y) : -\infty < x,y < \infty \}}$
2.: 例題7.5を用いて， $\displaystyle{\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} x^{-\frac{1}{2}}e^{-x} dx = \sqrt{\pi}}$ を示そう．