チェビシェフ多項式の別表記
チェビシェフ多項式の別表記
(1)
\[ T_{n}(x)=\frac{1}{2}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}+\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \]
(2)
\[ U_{n-1}(x)=\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}-\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \]
(1)
\begin{align*} T_{n}(x) & =\cos\left(n\cos^{\circ}x\right)\\ & =\frac{1}{2}\left(e^{in\cos^{\circ}x}+e^{-in\cos^{\circ}x}\right)\\ & =\frac{1}{2}\left(\left(e^{i\cos^{\circ}x}\right)^{n}+\left(e^{-i\cos^{\circ}x}\right)^{n}\right)\\ & =\frac{1}{2}\left(\left(\cos\cos^{\circ}x+i\sin\cos^{\circ}x\right)^{n}+\left(\cos\cos^{\circ}x-i\sin\cos^{\circ}x\right)^{n}\right)\\ & =\frac{1}{2}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}+\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \end{align*}
(2)
\begin{align*} U_{n-1}(x) & =\frac{\sin(n\cos^{\circ}x)}{\sin\cos^{\circ}x}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(e^{in\cos^{\circ}x}-e^{-in\cos^{\circ}x}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(\cos\cos^{\circ}x+i\sin\cos^{\circ}x\right)^{n}-\left(\cos\cos^{\circ}x-i\sin\cos^{\circ}x\right)^{n}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}-\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \end{align*}
ページ情報
タイトル | チェビシェフ多項式の別表記 |
URL | https://www.nomuramath.com/f28r2kv8/ |
SNSボタン |
- 第3種・第4種チェビシェフ多項式の直交性\[ \int_{-1}^{1}V_{m}(x)V_{n}(x)\sqrt{\frac{1+x}{1-x}}dx=\pi\delta_{mn} \]
- チェビシェフ多項式の級数表示\[ T_{n}(x)=\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(-1\right)^{k}\left(1-x^{2}\right)^{k}x^{n-2k}\right) \]
- チェビシェフ多項式の奇遇性\[ T_{n}(-x)=(-1)^{n}T_{n}(x) \]
- (*)チェビシェフ多項式のロドリゲス公式\[ T_{n}(x)=\frac{(-1)^{n}\sqrt{\pi}\sqrt{1-x^{2}}}{2^{n}\Gamma\left(n+\frac{1}{2}\right)}\frac{d^{n}}{dx^{n}}\left(1-x^{2}\right)^{n-\frac{1}{2}} \]