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Question Number 213564 by tri26112004 last updated on 09/Nov/24

$$f:Z\to Rsuchthat$$$$f\left(x\right).f\left(y\right)=f(x+y)+f(x-y)$$$$\Rightarrow f\left(x\right)=¿$$

Commented by mr W last updated on 09/Nov/24

$$f\left(x\right)=2\cos x$$

Commented by tri26112004 last updated on 09/Nov/24

$$yoursolution¿$$

Commented by mr W last updated on 09/Nov/24

$$i{don}^{\prime }tknowifthereareother$$$$solutions.$$

Commented by Ghisom last updated on 09/Nov/24

$$\mathrm{I}\mathrm{think}f\left(x\right)\mathrm{can}\mathrm{be}\mathrm{any}\mathrm{function}\mathrm{with}$$$$f\left(0\right)=2\wedge f\left(x\right)=f(-x)$$

Commented by mr W last updated on 09/Nov/24

$$ifyouwereright,thenf\left(x\right)=2+x^2$$$$isalsoasolution.butit{isn}^{\prime }t!$$

Commented by Ghisom last updated on 10/Nov/24

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{right}$$

Answered by issac last updated on 09/Nov/24

$$\mathrm{weyl}-\mathrm{wigner}\mathrm{transform}$$$$\mathrm{\Phi }\left[f(u,v)\right]=\frac{1}{4{\pi }^2}\underset{\mathcal{S}\in {\mathbb{R}}^2}{\int \int }\mathrm{d}u\mathrm{d}vf(u,v)e^{\mathit{i}uQ+\mathit{i}vP}$$$$\mathrm{set}\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$$$$\mathrm{and}\mathrm{double}\mathrm{integrals}\mathrm{interval}$$$${\mathbb{R}}^2\mathrm{mean}X\times Y=(-\mathrm{\infty },\mathrm{\infty })\times (-\mathrm{\infty }\mathrm{\infty })$$

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