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本帖最后由 青青子衿 于 2023-5-20 15:31 编辑
\begin{align*}
y^*&=\dfrac{1}{\mathcal{D}^2+1}\cos^3x=\operatorname{Re}\left(\dfrac{1}{\mathcal{D}^2+1}e^{\mathrm{i}x}\cos^2x\right)\\
&=\operatorname{Re}\left(e^{\mathrm{i}x}\dfrac{1}{\mathcal{D}^2+2\mathrm{i}\mathcal{D}}\cos^2x\right)\\
&=\operatorname{Re}\left(e^{\mathrm{i}x}\dfrac{1}{\mathcal{D}^2+2\mathrm{i}\mathcal{D}}\left(\dfrac{e^{\mathrm{i}x}}{2}\cos(x)+\dfrac{e^{-\mathrm{i}x}}{2}\cos(x)\right)\right)\\
&=\dfrac{1}{2}\operatorname{Re}\left(e^{\mathrm{i}x}e^{\mathrm{i}x}\dfrac{\cos(x)}{\mathcal{D}^2+4\mathrm{i}\mathcal{D}-3}+e^{\mathrm{i}x}e^{-\mathrm{i}x}\dfrac{\cos(x)}{\mathcal{D}^2+1}\right)\\
&=\dfrac{1}{2}\operatorname{Re}\left(e^{2\mathrm{i}x}\dfrac{1}{\mathcal{D}^2+4\mathrm{i}\mathcal{D}-3}\left(\dfrac{e^{\mathrm{i}x}+e^{-\mathrm{i}x}}{2}\right)\right)+\dfrac{1}{2}\dfrac{\cos(x)}{\mathcal{D}^2+1}\\
&=\dfrac{1}{4}\,\mathrm{Re}\left(e^{\mathrm{2i}x}\dfrac{1}{\mathcal{D}^2+4\mathrm{i}\mathcal{D}-3}e^{\mathrm{i}x}+
e^{\mathrm{2i}x}x\dfrac{1}{2\mathcal{D}+4\mathrm{i}}e^{\mathrm{-i}x}
\right)+\dfrac{x}{2}\dfrac{\cos(x)}{2\mathcal{D}} \\
&=\dfrac{1}{4}\,\mathrm{Re}\left(e^{\mathrm{2i}x}\dfrac{1}{\mathrm{i}^2+4\mathrm{i}\cdot\mathrm{i}-3}e^{\mathrm{i}x}+
e^{\mathrm{2i}x}x\dfrac{1}{2(-\mathrm{i})+4\mathrm{i}}e^{\mathrm{-i}x}
\right)+\dfrac{x}{4}\sin(x) \\
&=\dfrac{1}{4}\,\mathrm{Re}\left(-\dfrac{e^{\mathrm{3i}x}}{8}+
\dfrac{xe^{\mathrm{i}x}}{2\mathrm{i}}
\right)=\dfrac{1}{4}\left(-\dfrac{\cos(3x)}{8}+
\dfrac{x\sin(x)}{2}
\right) \\
&=-\dfrac{\cos(3x)}{32}+
\dfrac{x\sin(x)}{8}
\end{align*}
\begin{align*}
y^*&=\dfrac{1}{\mathcal{D}^2-2\mathrm{i}\mathcal{D}}1\\
&=\dfrac{1}{\mathcal{D}^2-2\mathrm{i}\mathcal{D}}\left(\dfrac{e^{\mathrm{i}x}}{2\mathrm{i}}\dfrac{1}{\sin\,\!x}-\dfrac{e^{-\mathrm{i}x}}{2\mathrm{i}}\dfrac{1}{\sin\,\!x}\right)\\
&=\dfrac{e^{\mathrm{i}x}}{2\mathrm{i}}\dfrac{1}{\mathcal{D}^2+1}\dfrac{1}{\sin\,\!x}-\dfrac{e^{-\mathrm{i}x}}{2\mathrm{i}}\dfrac{1}{\mathcal{D}^2-4\mathrm{i}\mathcal{D}-3}\dfrac{1}{\sin\,\!x}\\
\end{align*}
D[-(1/32) Cos[3 x] + 3/8 x Sin[x], x,
x] + (-(1/32) Cos[3 x] + 3/8 x Sin[x]) // FullSimplify
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