広田微分
基本的性質 広田微分は次の性質を満たす。 D x m f ⋅ 1 = ∂ x m f {\displaystyle D_{x}^{\,m}f\cdot 1=\partial _{x}^{\,m}f} D x m f ⋅ g = ( − 1 ) m D x m g ⋅ f {\displaystyle D_{x}^{\,m}f\cdot g=(-1)^{m}D_{x}^{\,m}g\cdot f} D x 2 m + 1 f ⋅ f = 0 {\displaystyle D_{x}^{\,2m+1}f\cdot f=0} D x m D t n e k 1 x − ω 1 t ⋅ e k 2 x − ω 2 t = ( k 1 − k 2 ) m ( − ω 1 + ω 2 ) n e ( k 1 x − ω 1 t ) + ( k 2 x − ω 2 t ) {\displaystyle D_{x}^{\,m}D_{t}^{\,n}e^{k_{1}x-\omega _{1}t}\cdot e^{k_{2}x-\omega _{2}t}=(k_{1}-k_{2})^{m}(-\omega _{1}+\omega _{2})^{n}e^{(k_{1}x-\omega _{1}t)+(k_{2}x-\omega _{2}t)}} F (Dt, Dx)をDt, Dxの多項式とすると F ( D t , D x ) e k 1 x − ω 1 t ⋅ e k 2 x − ω 2 t = F ( − ω 1 + ω 2 , k 1 − k 2 ) F ( − ω 1 − ω 2 , k 1 + k 2 ) F ( ∂ ∂ t , ∂ ∂ x ) e ( k 1 + k 2 ) x − ( ω 1 + ω 2 ) t {\displaystyle F(D_{t},D_{x})e^{k_{1}x-\omega _{1}t}\cdot e^{k_{2}x-\omega _{2}t}={\frac {F(-\omega _{1}+\omega _{2},k1-k_{2})}{F(-\omega _{1}-\omega _{2},k1+k_{2})}}F{\biggl (}{\frac {\partial }{\partial t}},\,{\frac {\partial }{\partial x}}{\biggr )}e^{(k1+k_{2})x-(\omega _{1}+\omega _{2})t}}