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ガンマ関数・ディガンマ関数を理解したい

Last updated at Posted at 2021-11-30

目標

  • 論文でガンマ・ディガンマ関数に遭遇してもひるまない自分を手に入れる

###ガンマ関数のモチベーション

  • 正の整数以外の数にも階乗を定義したい。(今回は正の実数への拡張のみを考える。)

定義

\Gamma(s) = \int_0^{\infty} x^{s-1}\space e^{-x}\space dx, \space s \in R^{+}

性質1

\Gamma(s) = (s-1)!,\space  s\in N

証明1

\Gamma(s) = (s-1)\Gamma(s-1)
= (s-1)(s-2)\Gamma(s-2)
=(s-1)(s-2)\cdots \Gamma(1)
= (s-1)(s-2)\cdots1
= (s-1)!

性質2

\Gamma(s) = (s-1)\Gamma(s-1), s>1

証明2

\Gamma(s) = \int_{0}^{\infty}x^{s-1} e^{-x}dx = [x^{s-1}-e^{-x}]_0^{\infty} - \int_0^{\infty}(s-1)x^{s-2}-e^{-x}dx= (s-1)\int_0^{\infty}x^{s-2}e^{-x}dx = (s-1)\Gamma(s-1)

性質3

\Gamma(\frac{1}{2}) = \sqrt \pi

証明3

\Gamma(\frac{1}{2}) = \int_0^{\infty} x^{-\frac{1}{2}}e^{-x} dx = \int_0^{\infty}t^{-1}e^{-t^2}2tdt = 2\int_0^{\infty} e^{-t^2}dt = \sqrt \pi

(t^2 = x \rightarrow dx/dt = 2t \rightarrow dx = 2tdt

\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt \pi)

視覚化

image.png

ディガンマ関数のモチベーション

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