基本的な関数の一般化超幾何関数表示
基本的な関数の一般化超幾何関数表示
基本的な関数を一般化超幾何関数\(F\left(a_{1},a_{2},\cdots,a_{m};b_{1},b_{2},\cdots,b_{n};x\right)\)を使って表示すると次のようになる。
基本的な関数を一般化超幾何関数\(F\left(a_{1},a_{2},\cdots,a_{m};b_{1},b_{2},\cdots,b_{n};x\right)\)を使って表示すると次のようになる。
(1)
\[ \left(1+x\right)^{a}=F\left(-a;;-x\right) \](2)
\[ e^{x}=F\left(;;x\right) \](3)
\[ \sin x=xF\left(;\frac{3}{2};-\frac{x^{2}}{4}\right) \](4)
\[ \cos x=F\left(;\frac{1}{2};-\frac{x^{2}}{4}\right) \](5)
\[ \log\left(1-x\right)=-xF\left(1,1;2;x\right) \](6)
\[ \log\frac{1-x}{1+x}=-2xF\left(\frac{1}{2},1;\frac{3}{2};x^{2}\right) \](7)
\[ \sin^{\bullet}x=xF\left(\frac{1}{2},\frac{1}{2};\frac{3}{2};x^{2}\right) \](8)
\[ \tan^{\bullet}x=xF\left(\frac{1}{2},1;\frac{3}{2};-x^{2}\right) \](1)
\begin{align*} \left(1+x\right)^{a} & =\sum_{k=0}^{\infty}P\left(a,k\right)\frac{x^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{k}Q\left(-a,k\right)\frac{x^{k}}{k!}\\ & =\sum_{k=0}^{\infty}Q\left(-a,k\right)\frac{\left(-x\right)^{k}}{k!}\\ & =F\left(-a;;-x\right) \end{align*}(2)
\begin{align*} e^{x} & =\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\\ & =F\left(;;x\right) \end{align*}(3)
\begin{align*} \sin x & =\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{\left(2k+1\right)!}x^{2k+1}\\ & =x\sum_{k=0}^{\infty}\frac{1}{\left(2k+1\right)!!\left(2k\right)!!}\left(-x^{2}\right)^{k}\\ & =x\sum_{k=0}^{\infty}\frac{1}{2^{k}Q\left(\frac{3}{2},k\right)2^{k}Q\left(1,k\right)}\left(-x^{2}\right)^{k}\\ & =x\sum_{k=0}^{\infty}\frac{1}{Q\left(\frac{3}{2},k\right)}\frac{\left(-\frac{x^{2}}{4}\right)^{k}}{k!}\\ & =xF\left(;\frac{3}{2};-\frac{x^{2}}{4}\right) \end{align*}(4)
\begin{align*} \cos x & =\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{\left(2k\right)!}x^{2k}\\ & =\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{\left(2k\right)!!\left(2k-1\right)!!}x^{2k}\\ & =\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{2^{k}k!2^{k}Q\left(\frac{1}{2},k\right)}x^{2k}\\ & =\sum_{k=0}^{\infty}\frac{1}{Q\left(\frac{1}{2},k\right)}\frac{\left(-\frac{x^{2}}{4}\right)^{k}}{k!}\\ & =F\left(;\frac{1}{2};-\frac{x^{2}}{4}\right) \end{align*}(5)
\begin{align*} \log\left(1-x\right) & =-\sum_{k=1}^{\infty}\frac{x^{k}}{k}\\ & =-\sum_{k=0}^{\infty}\frac{x^{k+1}}{k+1}\\ & =-x\sum_{k=0}^{\infty}\frac{k!k!}{\left(k+1\right)!}\frac{x^{k}}{k!}\\ & =-x\sum_{k=0}^{\infty}\frac{Q^{2}\left(1,k\right)}{Q\left(2,k\right)}\frac{x^{k}}{k!}\\ & =-xF\left(1,1;2;x\right) \end{align*}(6)
\begin{align*} \log\frac{1-x}{1+x} & =-xF\left(1,1;2;x\right)-xF\left(1,1;2;-x\right)\\ & =-x\sum_{k=0}^{\infty}\frac{Q^{2}\left(1,k\right)}{Q\left(2,k\right)}\frac{x^{k}+\left(-x\right)^{k}}{k!}\\ & =-2x\sum_{k=0}^{\infty}\frac{Q\left(1,2k\right)}{Q\left(2,2k\right)}x^{2k}\\ & =-2x\sum_{k=0}^{\infty}\frac{\left(2k\right)!}{\left(2k+1\right)!}x^{2k}\\ & =-2x\sum_{k=0}^{\infty}\frac{\left(2k\right)!!\left(2k-1\right)!!}{\left(2k+1\right)!!\left(2k\right)!!}x^{2k}\\ & =-2x\sum_{k=0}^{\infty}\frac{\left(2k-1\right)!!}{\left(2k+1\right)!!}x^{2k}\\ & =-2x\sum_{k=0}^{\infty}\frac{2^{k}Q\left(\frac{1}{2},k\right)}{2^{k}Q\left(\frac{3}{2},k\right)}x^{2k}\\ & =-2x\sum_{k=0}^{\infty}\frac{Q\left(\frac{1}{2},k\right)Q\left(1,k\right)}{Q\left(\frac{3}{2},k\right)}\frac{\left(x^{2}\right)^{k}}{k!}\\ & =-2xF\left(\frac{1}{2},1;\frac{3}{2};x^{2}\right) \end{align*}(7)
\begin{align*} \sin^{\bullet}x & =\sum_{k=0}^{\infty}\frac{\left(2k-1\right)!!}{(2k+1)\left(2k\right)!!}x^{2k+1}\\ & =\sum_{k=0}^{\infty}\frac{2^{k}Q\left(\frac{1}{2},k\right)}{2\left(k+\frac{1}{2}\right)2^{k}k!}x^{2k+1}\\ & =x\sum_{k=0}^{\infty}\frac{Q^{2}\left(\frac{1}{2},k\right)}{Q\left(\frac{3}{2},k\right)}\frac{\left(x^{2}\right)^{k}}{k!}\\ & =xF\left(\frac{1}{2},\frac{1}{2};\frac{3}{2};x^{2}\right) \end{align*}(8)
\begin{align*} \tan^{\bullet}x & =\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}x^{2k+1}\\ & =\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2\left(k+\frac{1}{2}\right)}x^{2k+1}\\ & =\sum_{k=0}^{\infty}\frac{(-1)^{k}Q\left(\frac{1}{2},k\right)}{Q\left(\frac{3}{2},k\right)}x^{2k+1}\\ & =x\sum_{k=0}^{\infty}\frac{Q\left(\frac{1}{2},k\right)Q\left(1,k\right)}{Q\left(\frac{3}{2},k\right)}\frac{\left(-x^{2}\right)^{k}}{k!}\\ & =xF\left(\frac{1}{2},1;\frac{3}{2};-x^{2}\right) \end{align*}
ページ情報
| タイトル | 基本的な関数の一般化超幾何関数表示 |
| URL | https://www.nomuramath.com/yjwmk1qe/ |
| SNSボタン |
一般化超幾何関数の微分と積分
\[
\frac{d}{dx}F\left(\boldsymbol{a};\boldsymbol{b};x\right)=\frac{\prod_{i=1}^{\dim\boldsymbol{a}}a_{i}}{\prod_{j=1}^{\dim\boldsymbol{b}}b_{j}}F\left(\boldsymbol{a}+\boldsymbol{1};\boldsymbol{b}+\boldsymbol{1};x\right)
\]
簡単な関数を上昇階乗とべき乗を使って表す
\[
ak+b=b\frac{Q\left(\frac{b}{a}+1,k\right)}{Q\left(\frac{b}{a},k\right)}
\]
幾何級数・超幾何級数・超幾何関数・合流型超幾何関数・一般化超幾何関数の定義
合流型超幾何微分方程式の解
\[
xy''\left(x\right)+\left(b-x\right)y'\left(x\right)-ay\left(x\right)=0
\]