【数学】莱布尼茨Leibniz公式求导
\(试证(a^2+b^2)^{\frac{n}{2}}e^{ax}sin(bx+n\phi)=\sum_{i=0}^{n}{\left(\begin{array}{c}n\\ i\end{array}\right)}a^{n-i}b^{i}e^{ax}sin(bx+\frac{\pi}{2}i),其中\phi = arctan(\frac{b}{a})\)
- \(令F(x)=e^{ax}sin(bx)\)
\[F^{'}(x)=ae^{ax}sin(bx)+be^{ax}cos(bx) \]
- \(合一变形\)
\[F^{'}(x)=e^{ax}(a^2+b^2)^{\frac{1}{2}}sin(bx+\phi) ,其中\phi = arctan(\frac{b}{a}) \]
- \(再重复n-1次求导\)
\[F^{(n)}(x)=e^{ax}(a^2+b^2)^{\frac{n}{2}}sin(bx+n\phi) \]
- \(另一方面,对F(x)应用Leibniz公式\)
\[F^{(n)}(x)=\sum_{i=0}^{n}{\left( \begin{array}{c}n \\ i\end{array}\right)a^{n-i}b^{i}e^{ax}sin(bx+\frac{\pi}{2}i)} \]
- \(得证\)