東京工業大学情報理工学院数理・計算科学系 2018年8月実施午前問B

Author

peter8rabit, 祭音Myyura

以下の問いに答えよ．ただし必要ならば

であることは証明なしで使ってよい．

(1) に対し，広義積分が収束することを示せ．

(2) に対し，ガンマ関数をと定める．このときを示せ．

(3) ガンマ関数のでの値を求めよ．

According to subadditivity of definite integrals, we split the integral into two as follows

For the first integral, we note that since , we can find a segment such that . Then, for and we have

By comparison test, we observe that the integral (1) converges.

For the second integral we note that

and again by comparison test, the integral (2) converges.