広義積分(improper integrals)

確認問題

1.

次の広義積分を求めよう．

(a) $${\iint_{\Omega}e^{y/x}dx dy, \Omega = \{(x,y) : 0 < x \leq 1, 0 \leq y \leq x \}}$$

(b) $${\iint_{\Omega}\frac{x}{\sqrt{1 - x - y}}dx dy, \Omega = \{(x,y) : x+y < 1, x \geq 0, y \geq 0\}}$$

(c) $${\iint_{\Omega}\log(x^{2} + y^{2})dxdy, \Omega = \{(x,y) : x,y \geq 0, x^2 + y^2 \leq 1 \}}$$

演習問題

1.

次の広義積分を求めよう．

(a) $${\iint_{\Omega}\frac{dxdy}{(y^2 - x^2)^{1/2}}, \Omega = \{(x,y) : 0 \leq x < y \leq 1\}}$$

(b) $${\iint_{\Omega}e^{-(x+y)}dxdy, \Omega = \{(x,y) : x \geq 0, y \geq 0\}}$$

(c) $${\iint_{\Omega}\tan^{-1}(\frac{y}{x})dxdy, \Omega = \{(x,y) : x,y \geq 0, x^2 + y^2 \leq 1 \}}$$

(d) $${\iint_{\Omega}\frac{1}{x^2y^{2}}dxdy, \Omega = \{(x,y) : x \geq 1, y \geq 1 \}}$$

(e) $${\iint_{\Omega}\frac{dxdy}{\sqrt{x - y^2}}, \Omega = \{(x,y) : 0 \leq x \leq 1, y^2 \leq x \}}$$

(f) $${\iint_{\Omega}\frac{dxdy}{1 + (x^2 + y^2)^2}, \Omega = \{(x,y) : -\infty < x,y < \infty \}}$$

2.

$\iint_{\Omega}e^{-(x^{2} + y^{2})}dx dy = \frac{\pi}{4}$を用いて， $${\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} x^{-\frac{1}{2}}e^{-x} dx = \sqrt{\pi}}$$ を示そう．