松原振動数の和の表
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2017/08/26 13:03 UTC 版)
以下の表に、 いくつかの簡単な有理関数g(z)での松原振動数の和をまとめる。 S η = 1 β ∑ i ω g ( i ω ) . {\displaystyle S_{\eta }={\frac {1}{\beta }}\sum _{i\omega }g(i\omega ).} η = ±1 は統計的記号である。 g ( i ω ) {\displaystyle g(i\omega )} S η {\displaystyle S_{\eta }} ( i ω − ξ ) − 1 {\displaystyle (i\omega -\xi )^{-1}} − η n η ( ξ ) {\displaystyle -\eta n_{\eta }(\xi )} [1] ( i ω − ξ ) − 2 {\displaystyle (i\omega -\xi )^{-2}} − η n η ′ ( ξ ) = β n η ( ξ ) ( η + n η ( ξ ) ) {\displaystyle -\eta n_{\eta }^{\prime }(\xi )=\beta n_{\eta }(\xi )(\eta +n_{\eta }(\xi ))} ( i ω − ξ ) − n {\displaystyle (i\omega -\xi )^{-n}} − η ( n − 1 ) ! ∂ ξ n − 1 n η ( ξ ) {\displaystyle -{\frac {\eta }{(n-1)!}}\partial _{\xi }^{n-1}n_{\eta }(\xi )} 1 ( i ω − ξ 1 ) ( i ω − ξ 2 ) {\displaystyle {\frac {1}{(i\omega -\xi _{1})(i\omega -\xi _{2})}}} − η ( n η ( ξ 1 ) − n η ( ξ 2 ) ) ξ 1 − ξ 2 {\displaystyle -{\frac {\eta (n_{\eta }(\xi _{1})-n_{\eta }(\xi _{2}))}{\xi _{1}-\xi _{2}}}} 1 ( i ω − ξ 1 ) 2 ( i ω − ξ 2 ) 2 {\displaystyle {\frac {1}{(i\omega -\xi _{1})^{2}(i\omega -\xi _{2})^{2}}}} η ( ξ 1 − ξ 2 ) 2 ( 2 ( n η ( ξ 1 ) − n η ( ξ 2 ) ) ξ 1 − ξ 2 − ( n η ′ ( ξ 1 ) + n η ′ ( ξ 2 ) ) ) {\displaystyle {\frac {\eta }{(\xi _{1}-\xi _{2})^{2}}}\left({\frac {2(n_{\eta }(\xi _{1})-n_{\eta }(\xi _{2}))}{\xi _{1}-\xi _{2}}}-(n_{\eta }^{\prime }(\xi _{1})+n_{\eta }^{\prime }(\xi _{2}))\right)} 1 ( i ω − ξ 1 ) 2 − ξ 2 2 {\displaystyle {\frac {1}{(i\omega -\xi _{1})^{2}-\xi _{2}^{2}}}} η c η ( ξ 1 , ξ 2 ) {\displaystyle \eta c_{\eta }(\xi _{1},\xi _{2})} 1 ( i ω ) 2 − ξ 2 {\displaystyle {\frac {1}{(i\omega )^{2}-\xi ^{2}}}} η c η ( 0 , ξ ) = − 1 2 ξ ( 1 + 2 η n η ( ξ ) ) {\displaystyle \eta c_{\eta }(0,\xi )=-{\frac {1}{2\xi }}(1+2\eta n_{\eta }(\xi ))} ( i ω ) 2 ( i ω ) 2 − ξ 2 {\displaystyle {\frac {(i\omega )^{2}}{(i\omega )^{2}-\xi ^{2}}}} − ξ 2 ( 1 + 2 η n η ( ξ ) ) {\displaystyle -{\frac {\xi }{2}}(1+2\eta n_{\eta }(\xi ))} 1 ( ( i ω ) 2 − ξ 2 ) 2 {\displaystyle {\frac {1}{((i\omega )^{2}-\xi ^{2})^{2}}}} − η 2 ξ 2 ( c η ( 0 , ξ ) + n η ′ ( ξ ) ) {\displaystyle -{\frac {\eta }{2\xi ^{2}}}(c_{\eta }(0,\xi )+n_{\eta }^{\prime }(\xi ))} ( i ω ) 2 ( ( i ω ) 2 − ξ 2 ) 2 {\displaystyle {\frac {(i\omega )^{2}}{((i\omega )^{2}-\xi ^{2})^{2}}}} η 2 ( c η ( 0 , ξ ) − n η ′ ( ξ ) ) {\displaystyle {\frac {\eta }{2}}(c_{\eta }(0,\xi )-n_{\eta }^{\prime }(\xi ))} ( i ω ) 2 + ξ 2 ( ( i ω ) 2 − ξ 2 ) 2 {\displaystyle {\frac {(i\omega )^{2}+\xi ^{2}}{((i\omega )^{2}-\xi ^{2})^{2}}}} − η n η ′ ( ξ ) = β n η ( ξ ) ( η + n η ( ξ ) ) {\displaystyle -\eta n_{\eta }^{\prime }(\xi )=\beta n_{\eta }(\xi )(\eta +n_{\eta }(\xi ))} 1 ( ( i ω ) 2 − ξ 1 2 ) ( ( i ω ) 2 − ξ 2 2 ) {\displaystyle {\frac {1}{((i\omega )^{2}-\xi _{1}^{2})((i\omega )^{2}-\xi _{2}^{2})}}} η ( c η ( 0 , ξ 1 ) − c η ( 0 , ξ 2 ) ) ξ 1 2 − ξ 2 2 {\displaystyle {\frac {\eta (c_{\eta }(0,\xi _{1})-c_{\eta }(0,\xi _{2}))}{\xi _{1}^{2}-\xi _{2}^{2}}}} ( 1 ( i ω ) 2 − ξ 1 2 + 1 ( i ω ) 2 − ξ 2 2 ) 2 {\displaystyle \left({\frac {1}{(i\omega )^{2}-\xi _{1}^{2}}}+{\frac {1}{(i\omega )^{2}-\xi _{2}^{2}}}\right)^{2}} η ( 3 ξ 1 2 + ξ 2 2 2 ξ 1 2 ( ξ 1 2 − ξ 2 2 ) c η ( 0 , ξ 1 ) − n η ′ ( ξ 1 ) 2 ξ 1 2 ) + ( 1 ↔ 2 ) {\displaystyle \eta \left({\frac {3\xi _{1}^{2}+\xi _{2}^{2}}{2\xi _{1}^{2}(\xi _{1}^{2}-\xi _{2}^{2})}}c_{\eta }(0,\xi _{1})-{\frac {n_{\eta }^{\prime }(\xi _{1})}{2\xi _{1}^{2}}}\right)+(1\leftrightarrow 2)} [2] ( 1 ( i ω ) 2 − ξ 1 2 − 1 ( i ω ) 2 − ξ 2 2 ) 2 {\displaystyle \left({\frac {1}{(i\omega )^{2}-\xi _{1}^{2}}}-{\frac {1}{(i\omega )^{2}-\xi _{2}^{2}}}\right)^{2}} η ( − 5 ξ 1 2 − ξ 2 2 2 ξ 1 2 ( ξ 1 2 − ξ 2 2 ) c η ( 0 , ξ 1 ) − n η ′ ( ξ 1 ) 2 ξ 1 2 ) + ( 1 ↔ 2 ) {\displaystyle \eta \left(-{\frac {5\xi _{1}^{2}-\xi _{2}^{2}}{2\xi _{1}^{2}(\xi _{1}^{2}-\xi _{2}^{2})}}c_{\eta }(0,\xi _{1})-{\frac {n_{\eta }^{\prime }(\xi _{1})}{2\xi _{1}^{2}}}\right)+(1\leftrightarrow 2)} [2] [1] 和は収束しないため、松原重み関数の選択が異なると結果は定数分だけ異なる。 [2] (1 ↔ 2)は、手前の項のインデックス１と２を置き換えたものを表す。
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