Question Number 213564 by tri26112004 last updated on 09/Nov/24
$${f}:\:{Z}\rightarrow{R}\:{such}\:{that} \\ $$$${f}\left({x}\right).{f}\left({y}\right)={f}\left({x}+{y}\right)+{f}\left({x}−{y}\right) \\ $$$$\Rightarrow{f}\left({x}\right)=¿ \\ $$
Commented by mr W last updated on 09/Nov/24
$${f}\left({x}\right)=\mathrm{2}\:\mathrm{cos}\:{x} \\ $$
Commented by tri26112004 last updated on 09/Nov/24
$${your}\:{solution}¿ \\ $$
Commented by mr W last updated on 09/Nov/24
$${i}\:{don}'{t}\:{know}\:{if}\:{there}\:{are}\:{other}\: \\ $$$${solutions}. \\ $$
Commented by Ghisom last updated on 09/Nov/24
$$\mathrm{I}\:\mathrm{think}\:{f}\left({x}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{any}\:\mathrm{function}\:\mathrm{with} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{2}\wedge{f}\left({x}\right)={f}\left(−{x}\right) \\ $$
Commented by mr W last updated on 09/Nov/24
$${if}\:{you}\:{were}\:{right},\:{then}\:{f}\left({x}\right)=\mathrm{2}+{x}^{\mathrm{2}} \\ $$$${is}\:{also}\:{a}\:{solution}.\:{but}\:{it}\:{isn}'{t}! \\ $$
Commented by Ghisom last updated on 10/Nov/24
$$\mathrm{you}'\mathrm{re}\:\mathrm{right} \\ $$
Answered by issac last updated on 09/Nov/24
![weyl−wigner transform Φ[f(u,v)]=(1/(4π^2 ))∫∫_(S∈R^2 ) dudv f(u,v)e^(iuQ+ivP) set R=(−∞,∞) and double integrals interval R^2 mean X×Y=(−∞,∞)×(−∞∞)](https://www.tinkutara.com/question/Q213570.png)
$$\mathrm{weyl}−\mathrm{wigner}\:\mathrm{transform} \\ $$$$\Phi\left[{f}\left({u},{v}\right)\right]=\frac{\mathrm{1}}{\mathrm{4}\pi^{\mathrm{2}} }\underset{\mathcal{S}\in\mathbb{R}^{\mathrm{2}} } {\int\int}\:\:\mathrm{d}{u}\mathrm{d}{v}\:{f}\left({u},{v}\right){e}^{\boldsymbol{{i}}{uQ}+\boldsymbol{{i}}{vP}} \\ $$$$\mathrm{set}\:\mathbb{R}=\left(−\infty,\infty\right) \\ $$$$\mathrm{and}\:\mathrm{double}\:\mathrm{integrals}\:\mathrm{interval} \\ $$$$\mathbb{R}^{\mathrm{2}} \:\mathrm{mean}\:{X}×{Y}=\left(−\infty,\infty\right)×\left(−\infty\infty\right) \\ $$