5.4 Improper integrals

1.

(a) $${\frac{\pi}{2}}$$(b) 1(c) $${\frac{\pi^2}{16}}$$(d) 1(e) $\pi$(f) $${\frac{\pi^2}{2}}$$

2.

Let $${t = x^{\frac{1}{2}}}$$. Then $${2tdt = dx}$$．Thus，

$$\Gamma(\frac{1}{2}) = \int_{0}^{\infty} x^{-\frac{1}{2}}e^{-x} dx = 2\int_{0}^{\infty} e^{-t^2} dt$$

Let $${I = \int_{0}^{\infty} e^{-x^2} dx, I = \int_{0}^{\infty} e^{-y^2} dt}$$. Then

$$I^2 = \iint_{0}^{\infty} e^{-(x^2 + y^2)} dx dy$$

Note that by 5.5， $${I^2 = \frac{\pi}{4}}$$ より $${\Gamma(\frac{1}{2}) = \sqrt{\pi}}$$