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^{\bullet}x\right)\\ & =\frac{1}{2}\left(e^{in\cos^{\bullet}x}+e^{-in\cos^{\bullet}x}\right)\\ & =\frac{1}{2}\left(\left(e^{i\cos^{\bullet}x}\right)^{n}+\left(e^{-i\cos^{\bullet}x}\right)^{n}\right)\\ & =\frac{1}{2}\left(\left(\cos\cos^{\bullet}x+i\sin\cos^{\bullet}x\right)^{n}+\left(\cos\cos^{\bullet}x-i\sin\cos^{\bullet}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^{\bullet}x)}{\sin\cos^{\bullet}x}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(e^{in\cos^{\bullet}x}-e^{-in\cos^{\bullet}x}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(\cos\cos^{\bullet}x+i\sin\cos^{\bullet}x\right)^{n}-\left(\cos\cos^{\bullet}x-i\sin\cos^{\bullet}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 |