\(\cos{\left(\left(m+1\right)t\right)}=\cos{\left(t\right)}\cos{\left(mt\right)}-\sin{\left(t\right)}\sin{\left(mt\right)}\\
\cos{\left(\left(m-1\right)t\right)}=\cos{\left(t\right)}\cos{\left(mt\right)}+\sin{\left(t\right)}\sin{\left(mt\right)}\\
\cos{\left(\left(m+1\right)t\right)}+\cos{\left(\left(m-1\right)t\right)}=2\cos{\left(t\right)}\cos{\left(mt\right)}\\
T_{m+1} \left( x\right) +T_{m-1} \left(x \right) = 2xT_{m} \left(x \right) \\
T_{m+1} \left( x\right) = 2xT_{m} \left(x \right) - T_{m-1} \left(x \right) \\
m+1 = n\\
T_{n} \left( x\right) = 2xT_{n-1} \left(x \right) - T_{n-2} \left(x \right) \\
\begin{cases} T_{n} \left(x \right) = 1 \qquad n = 0 \\ T_{n} \left(x \right) = x \qquad n = 1 \\ T_{n} \left( x\right) = 2xT_{n-1} \left(x \right) - T_{n-2} \left(x \right) \qquad n \ge 2\end{cases}
\)
\(
E \left(x,t \right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!}\\
\sum_{n=2}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!} = \sum_{n=2}^{ \infty }2xT_{n-1} \left(x \right) \frac{t^n}{n!} + \sum_{n=2}^{ \infty }\left(-T_{n-2} \left(x \right)\right) \frac{t^n}{n!}\\
\sum_{n=2}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!} = 2x\left(\sum_{n=2}^{ \infty }T_{n-1} \left(x \right) \frac{t^n}{n!}\right) -\left(\sum_{n=2}^{ \infty }T_{n-2} \left(x \right) \frac{t^n}{n!}\right)\\
\sum_{n=2}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!} = 2x\left(\sum_{n=1}^{ \infty }T_{n} \left(x \right) \frac{t^{n+1}}{\left(n+1\right)!}\right) -\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+2}}{\left(n+2\right)!}\right)\\
\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!}-1-xt = 2x\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+1}}{\left(n+1\right)!} - t\right) -\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+2}}{\left(n+2\right)!}\right)\\
\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!}-2x\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+1}}{\left(n+1\right)!}\right)+\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+2}}{\left(n+2\right)!}\right)=1+xt-2xt\\
\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^n}{n!}-2x\left(\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+1}}{\left(n+1\right)!}\right)+\sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+2}}{\left(n+2\right)!}=1-xt\\
P\left(t\right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+2}}{\left(n+2\right)!}\\
P'\left(t\right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{\left(n+2\right)t^{n+1}}{\left(n+2\right)\left(n+1\right)!}\\
P'\left(t\right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n+1}}{\left(n+1\right)!}\\
P''\left(t\right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{\left(n+1\right)t^{n}}{\left(n+1\right)n!}\\
P''\left(t\right) = \sum_{n=0}^{ \infty }T_{n} \left(x \right) \frac{t^{n}}{n!}\\
\begin{cases}P'' \left(t\right) - 2xP'\left(t\right)+P\left(t\right) = 1-xt \\ P \left( 0 \right) = 0\\P' \left( 0\right)=0 \end{cases} \\
\)
\(\mathcal{L} \{P'' \left(t \right) - 2xP' \left( t\right) + P \left( t\right) \}=\mathcal{L} \{1-xt \} = \\
\mathcal{L} \{P'' \left(t \right) \} - 2x\mathcal{L}\{P' \left( t\right)\} +\mathcal{L} \{P \left( t\right) \} = \mathcal{L} \{1\} -x\mathcal{L}\{t\}\\
\left( -P' \left(0 \right)-P \left( 0\right)s + s^2P \left(s \right) \right)-2x \left(-P \left(0 \right) +sP \left(s \right) \right)+P \left(s \right)=\frac{1}{s} -\frac{x}{s^2} \\
s^2P \left(s \right)-2xsP \left(s \right)+P \left(s \right)=\frac{s-x}{s^2}\\
\left(s^2-2xs+1\right)P \left(s\right)=\frac{s-x}{s^2}\\
P \left(s\right) = \frac{s-x}{s^2\left(s^2-2xs+1\right)}\\
\frac{s-x}{s^2\left(s^2-2xs+1\right)} = \frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2-2xs+1}\\
s-x = As \left(s^2-2xs+1 \right) +B \left(s^2-2xs+1 \right) + \left(Cs+D \right)s^2\\
s-x = A\left(s^3-2xs^2+s \right)+B \left(s^2-2xs+1 \right)+Cs^3+Ds^2\\
s-x =\left(A+C\right)s^3+\left(-2xA+B+D\right)s^2+\left(A-2xB\right)s+B\\
\begin{cases}A+C = 0\\-2xA+B+D = 0\\ A-2xB = 1\\B = -x \end{cases} \\
\begin{cases}C = -A\\-2x \left(1-2x^2 \right) -x+D = 0\\ A = 1 - 2x^2\\B = -x \end{cases} \\
\begin{cases}C = 2x^2-1\\D = 3x - 4x^3\\ A = 1 - 2x^2\\B = -x \end{cases} \\
P \left( t\right) = \mathcal{L}^{-1} \{\frac{s-x}{s^2\left(s^2-2xs+1 \right)} \} \\
P \left( t\right) = \mathcal{L}^{-1} \{ \left(1-2x^2 \right) \cdot \frac{1}{s} -x \cdot \frac{1}{s^2}+\frac{ \left(2x^2-1 \right)s+3x-4x^3}{s^2-2xs+1}\}\\
P \left( t\right) = \left(1-2x^2 \right)\mathcal{L}^{-1} \{\frac{1}{s}\} -x\mathcal{L}^{-1} \{\frac{1}{s^2}\}+\mathcal{L}^{-1}\{\frac{ \left(2x^2-1 \right)s+3x-4x^3}{s^2-2xs+1}\}\\
P \left( t\right) = \left(1-2x^2 \right)\mathcal{L}^{-1} \{\frac{1}{s}\} -x\mathcal{L}^{-1} \{\frac{1}{s^2}\}+\mathcal{L}^{-1}\{\frac{ \left(2x^2-1 \right)\left(s-x\right)+2x^3-x+3x-4x^3}{\left(s-x\right)^2+1-x^2}\}\\
P \left( t\right) = \left(1-2x^2 \right)\mathcal{L}^{-1} \{\frac{1}{s}\} -x\mathcal{L}^{-1} \{\frac{1}{s^2}\}+\mathcal{L}^{-1}\{\frac{ \left(2x^2-1 \right)\left(s-x\right)+2x-2x^3}{\left(s-x\right)^2+1-x^2}\}\\
P \left( t\right) = \left(1-2x^2 \right)\mathcal{L}^{-1} \{\frac{1}{s}\} -x\mathcal{L}^{-1} \{\frac{1}{s^2}\}+\left(2x^2-1 \right)\mathcal{L}^{-1}\{\frac{ s-x}{\left(s-x\right)^2+1-x^2}\} +\frac{2x-2x^3}{\sqrt{1-x^2}} \mathcal{L}^{-1}\{\frac{\sqrt{1-x^2}}{\left(s-x\right)^2+1-x^2}\}\\
P \left( t\right) = \left(1-2x^2 \right)\mathcal{L}^{-1} \{\frac{1}{s}\} -x\mathcal{L}^{-1} \{\frac{1}{s^2}\}+\left(2x^2-1 \right)\mathcal{L}^{-1}\{\frac{ s-x}{\left(s-x\right)^2+1-x^2}\} +2x\sqrt{1-x^2} \mathcal{L}^{-1}\{\frac{\sqrt{1-x^2}}{\left(s-x\right)^2+1-x^2}\}\\
P \left( t\right) = \left(1-2x^2 \right) - xt + \left(2x^2-1 \right)e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}+2x\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}
\)
\(
P \left( t\right) = \left(1-2x^2 \right) - xt + \left(2x^2-1 \right)e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}+2x\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\\
P'\left( t\right) = -x + \left(2x^2-1 \right)\left(xe^{xt}\cos{\left(t\sqrt{1-x^2}\right)}-\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\right)+2x\sqrt{1-x^2}\left(xe^{xt}\sin{\left(t\sqrt{1-x^2}\right)}+\sqrt{1-x^2}e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}\right)\\
P'\left( t\right) = -x +\left(2x^3-x \right)e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}+\left(2x-2x^3\right)e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}-\left(2x^2-1\right)\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}+2x^2\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\\
P'\left( t\right) = -x + xe^{xt}\cos{\left(t\sqrt{1-x^2}\right)}+\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\\
P''\left( t\right) = x\left(xe^{xt}\cos{\left(t\sqrt{1-x^2}\right)}-\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\right)+\sqrt{1-x^2}\left(xe^{xt}\sin{\left(t\sqrt{1-x^2}\right)}+\sqrt{1-x^2}e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}\right)\\
P''\left( t\right) = x^2e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}+\left(1-x^2\right)e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}-x\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}+x\sqrt{1-x^2}e^{xt}\sin{\left(t\sqrt{1-x^2}\right)}\\
P''\left( t\right) = e^{xt}\cos{\left(t\sqrt{1-x^2}\right)} \\
E\left(x,t\right) = e^{xt}\cos{\left(t\sqrt{1-x^2}\right)}\\
\)
\(
\left(fg\right)^{ \left( n\right) }=\sum_{k=0}^{n} { n \choose k} f^{ \left(k \right) }g^{ \left(n-k \right) }\\
\frac{\mbox{d}^n}{\mbox{d}t^n} e^{xt} = x^{n}e^{xt}\\
\frac{\mbox{d}^n}{\mbox{d}t^n} \cos{\left(t\sqrt{1-x^2}\right)} = \left( \sqrt{1-x^2} \right) ^{n}\cos{ \left(\frac{\pi}{2}n + t\sqrt{1-x^2} \right) }\\
T_{n}\left(x\right)=\sum_{k=0}^{n} { n \choose k} x^{n-k}\left( \sqrt{1-x^2} \right) ^{k}\cos{ \left(\frac{\pi}{2}k \right) }\\
T_{n}\left(x\right)=\sum_{k=0}^{\lfloor \frac{n}{2}\rfloor} { n \choose 2k} \left(-1\right)^kx^{n-2k}\left( 1 - x^2 \right) ^{k}\\
\)
Wielomiany Czebyszowa
Otrzymałeś(aś) rozwiązanie do zamieszczonego zadania? - podziękuj autorowi rozwiązania! Kliknij