Mathématiques | Texte intégral gratuit | Une revue des distributions enveloppées pour les données circulaires

$$\begin{array}{c}\hfill {\displaystyle F\left(\theta \right)=\mathrm{UN}{\mathrm{et}}^{-\lambda \theta }\left({\theta }^3+B{\theta }^2+C\theta +D\right)}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F\left(\theta \right)=\frac{\mathrm{UN}}{{\lambda }^4}\left[\gamma (4,\lambda \theta )+B\lambda \gamma (3,\lambda \theta )+C{\lambda }^2\gamma (2,\lambda \theta )+D{\lambda }^3\gamma (1,\lambda \theta )\right]}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \mathrm{UN}=\frac{{\lambda }^4}{\left({\lambda }^3+6\right)\left(1-{\mathrm{et}}^{-2\lambda \pi }\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle B=\frac{6\pi {\mathrm{et}}^{-2\lambda \pi }}{1-{\mathrm{et}}^{-2\lambda \pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle C=\frac{2\pi \left({\mathrm{et}}^{-2\lambda \pi }+1\right)B}{1-{\mathrm{et}}^{-2\lambda \pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle D=1+\frac{8{\pi }^3{\mathrm{et}}^{-2\lambda \pi }\left({\mathrm{et}}^{-4\lambda \pi }+4{\mathrm{et}}^{-2\lambda \pi }+1\right)}{{\left(1-{\mathrm{et}}^{-2\lambda \pi }\right)}^3}.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle m_n=\frac{\mathrm{UN}}{{\left({\lambda }^2+n^2\right)}^4}\left[\left(c_0+c_2{n}^2+c_4{n}^4+c_6{n}^6\right)+\mathrm{je}\left(d_{1}n+d_3{n}^2+d_5{n}^5+d_7{n}^7\right)\right]}\end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle c_0={\lambda }^7-4\frac{{\lambda }^{6}B\pi }{{\left({\mathrm{et}}^{\pi \lambda }\right)}^2}-4\frac{{\lambda }^{7}B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}-2\frac{{\lambda }^{7}C\pi }{{\mathrm{et}}^{2\pi \lambda }}-2\frac{{\lambda }^{5}B}{{\mathrm{et}}^{2\pi \lambda }}-\frac{{\lambda }^{6}C}{{\mathrm{et}}^{2\pi \lambda }}-6\frac{{\lambda }^4}{{\mathrm{et}}^{2\pi \lambda }}-12\frac{{\pi }^2{\lambda }^6}{{\mathrm{et}}^{2\pi \lambda }}}\hfill \\ \hfill & {\displaystyle -8\frac{{\pi }^3{\lambda }^7}{{\mathrm{et}}^{2\pi \lambda }}-12\frac{\pi {\lambda }^5}{{\mathrm{et}}^{2\pi \lambda }}+6{\lambda }^4+{\lambda }^{6}C-\frac{{\lambda }^7}{{\mathrm{et}}^{2\pi \lambda }}+2{\lambda }^{5}B,}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle c_2=-\frac{{\lambda }^{4}C}{{\mathrm{et}}^{2\pi \lambda }}+4\frac{{\lambda }^{3}B}{{\mathrm{et}}^{2\pi \lambda }}+24\frac{\pi {\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}-12\frac{{\pi }^2{\lambda }^4}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^3{\lambda }^5}{{\mathrm{et}}^{2\pi \lambda }}+{\lambda }^{4}C-3\frac{{\lambda }^5}{{\mathrm{et}}^{2\pi \lambda }}-4{\lambda }^{3}B-36{\lambda }^2}\hfill \\ \hfill & {\displaystyle +36\frac{{\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }}+3{\lambda }^5-4\frac{{\lambda }^{4}B\pi }{{\mathrm{et}}^{2\pi \lambda }}-6\frac{{\lambda }^{5}C\pi }{{\mathrm{et}}^{2\pi \lambda }}-12\frac{{\lambda }^{5}B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }},}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle c_4=\frac{{\lambda }^{2}C}{{\mathrm{et}}^{2\pi \lambda }}+6\frac{\lambda B}{{\mathrm{et}}^{2\pi \lambda }}+36\frac{\pi \lambda }{{\mathrm{et}}^{2\pi \lambda }}+12\frac{{\pi }^2{\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^3{\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}-3\frac{{\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}-{\lambda }^{2}C-6\lambda B-\frac{6}{{\mathrm{et}}^{2\pi \lambda }}}\hfill \\ \hfill & {\displaystyle +6+3{\lambda }^3-6\frac{C\pi {\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}-12\frac{B{\pi }^2{\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}+4\frac{B\pi {\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle c_6=4\frac{B\pi }{{\mathrm{et}}^{2\pi \lambda }}-8\frac{{\pi }^3\lambda }{{\mathrm{et}}^{2\pi \lambda }}-\frac{\lambda }{{\mathrm{et}}^{2\pi \lambda }}+\frac{C}{{\mathrm{et}}^{2\pi \lambda }}+12\frac{{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}+\lambda -C-2\frac{\lambda C\pi }{{\mathrm{et}}^{2\pi \lambda }}-4\frac{\lambda B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }},}\end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle d_1=2{\lambda }^{5}C-\frac{{\lambda }^6}{{\mathrm{et}}^{2\pi \lambda }}+6{\lambda }^{4}B-6\frac{{\lambda }^{4}B}{{\mathrm{et}}^{2\pi \lambda }}-2\frac{{\lambda }^{5}C}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^2{\lambda }^5}{{\mathrm{et}}^{2\pi \lambda }}-8\frac{{\pi }^3{\lambda }^6}{{\mathrm{et}}^{2\pi \lambda }}-3\frac{\pi {\lambda }^4}{{\mathrm{et}}^{2\pi \lambda }}+24{\lambda }^3}\hfill \\ \hfill & {\displaystyle -24\frac{{\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}+{\lambda }^6-8\frac{{\lambda }^{5}B\pi }{{\mathrm{et}}^{2\pi \lambda }}-4\frac{{\lambda }^{6}B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}-2\frac{{\lambda }^{6}C\pi }{{\mathrm{et}}^{2\pi \lambda }},}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle d_3=-3\frac{{\lambda }^4}{{\mathrm{et}}^{2\pi \lambda }}+4{\lambda }^{3}C+4{\lambda }^{2}B-4\frac{{\lambda }^{3}C}{{\mathrm{et}}^{2\pi \lambda }}-4\frac{{\lambda }^{2}B}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{\pi {\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }}-48\frac{{\pi }^2{\lambda }^3}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^3{\lambda }^4}{{\mathrm{et}}^{2\pi \lambda }}-24\lambda }\hfill \\ \hfill & {\displaystyle +24\frac{\lambda }{{\mathrm{et}}^{2\pi \lambda }}+3{\lambda }^4-6\frac{{\lambda }^{4}C\pi }{{\mathrm{et}}^{2\pi \lambda }}-12\frac{{\lambda }^{4}B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}-16\frac{{\lambda }^{3}B\pi }{{\mathrm{et}}^{2\pi \lambda }},}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle d_5=2\lambda C+2\frac{B}{{\mathrm{et}}^{2\pi \lambda }}-3\frac{{\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }}-2\frac{\lambda C}{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^2\lambda }{{\mathrm{et}}^{2\pi \lambda }}-24\frac{{\pi }^3{\lambda }^2}{{\mathrm{et}}^{2\pi \lambda }}+12\frac{\pi }{{\mathrm{et}}^{2\pi \lambda }}+3{\lambda }^2-2B}\hfill \\ \hfill & {\displaystyle -6\frac{{\lambda }^{2}C\pi }{{\mathrm{et}}^{2\pi \lambda }}-12\frac{{\lambda }^{2}B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}-8\frac{\lambda B\pi }{{\mathrm{et}}^{2\pi \lambda }},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle d_7=1-2\frac{C\pi }{{\mathrm{et}}^{2\pi \lambda }}-4\frac{B{\pi }^2}{{\mathrm{et}}^{2\pi \lambda }}-8\frac{{\pi }^3}{{\mathrm{et}}^{2\pi \lambda }}-\frac{1}{{\mathrm{et}}^{2\pi \lambda }}.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \mu =arcsin\left[\frac{d_1+d_3+d_5+d_7}{\sqrt{{\left(c_0+c_2+c_4+c_6\right)}^2+{\left(d_1+d_3+d_5+d_7\right)}^2}}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \rho =\frac{\mathrm{UN}}{{\left({\lambda }^2+1\right)}^4}\sqrt{{\left(c_0+c_2+c_4+c_6\right)}^2+{\left(d_1+d_3+d_5+d_7\right)}^2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle {\gamma }_1=\frac{\mathrm{UN}{\mathrm{et}}^{-2\mu }}{v^{\frac{3}{2}}{\left({\lambda }^2+4\right)}^4}\left(2d_1+4d_3+32d_5+128d_7\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle {\gamma }_2=\frac{\mathrm{UN}{\mathrm{et}}^{-2\mu }}{\left(1-{\rho }^2\right){\left({\lambda }^2+4\right)}^4}\left(c_0+4c_2+16c_4+64c_6\right)-\frac{{\rho }^4}{1-{\rho }^2},}\end{array}$$