γ = √((R+jωL)(G+jωC)) = √(RG-LCω^2+jω(LG+CR)) = ω√(LC)√(RG/LCω^2-1+j(G/ωC+R/ωL))

α = ω√(LC)         √[{   a+√(a^2+b^2)                                                                         }/2] :複素数の平方根(*rem1)
α = ω√(LC)         √[{   RG/LCω^2 - 1 + √((RG/LCω^2-1)^2+(G/ωC+R/ωL)^2)                                       }/2]
α = ω√(LC)         √[{   RG/LCω^2 - 1 + √((RG/LCω^2-1)^2+(G/ωC+R/ωL)^2)                                       }/2] 
α = ω√(LC)         √[{   RG/LCω^2 - 1 + √((RG/LCω^2)^2 + 1 - 2(RG/LCω^2) + (G/ωC)^2 + (R/ωL)^2 + 2(GR/LCω^2)) }/2]
α = ω√(LC)         √[{   RG/LCω^2 - 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 )                           }/2] 
α = ω√(LC)         √[{   RG/LCω^2 - 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 )                           }/2]

β = ω√(LC)         √[{  -a+√(a^2+b^2)                                                                         }/2] :複素数の平方根(*rem1)
β = ω√(LC)         √[{  -RG/LCω^2 + 1 + √((RG/LCω^2-1)^2+(G/ωC+R/ωL)^2)                                       }/2]
β = ω√(LC)         √[{  -RG/LCω^2 + 1 + √((RG/LCω^2-1)^2+(G/ωC+R/ωL)^2)                                       }/2] 
β = ω√(LC)         √[{  -RG/LCω^2 + 1 + √((RG/LCω^2)^2 + 1 - 2(RG/LCω^2) + (G/ωC)^2 + (R/ωL)^2 + 2(GR/LCω^2)) }/2]
β = ω√(LC)         √[{  -RG/LCω^2 + 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 )                           }/2] 
β = ω√(LC)         √[{  -RG/LCω^2 + 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 )                           }/2]





【低周波領域】
α = ω√(LC) √[{ RG/LCω^2 - 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 ) }/2] α = ω√(LC) √[{ RG/LCω^2 - 1 + 1/ω^2 √( ω^4 + (RG/LC)^2 + (ωG/C)^2 + (ωR/L)^2 ) }/2] α = ω√(LC) √[{ RG/LCω^2 - 1 + (RG/LC)/ω^2 √( ω^4/(RG/LC)^2 + 1 + (ωL/R)^2 + (ωC/G)^2 ) }/2] α = ω√(LC) √[{ RG/LCω^2 - 1 + (RG/LC)/ω^2 √( 1 + (ωL/R)^2 + (ωC/G)^2 ) }/2] : ωL/R, ωC/G の4次項を削除 α = ω√(LC) √[{ RG/LCω^2 - 1 + (RG/LC)/ω^2  ( 1 + (ωL/R)^2/2 + (ωC/G)^2/2 ) }/2] : 2項式の1/2乗近似(1との比較) α = ω√(LC)   √[{ 2RG/LCω^2 - 1 + (LG/RC)/2 + (CR/LG)/2 }/2] α = ω√(LC)   √[ RG/LCω^2 - 1/2 + (LG/RC)/4 + (CR/LG)/4 ] α = ω√(LC) (1/ω) √[ RG/LC + ω^2( -1/2 + (LG/RC)/4 + (CR/LG)/4 ) ] α = ω√(LC) √(RG/LC)/ω √[ 1 + ω^2( -(LC/RG)/2 + (L/R)^2/4 + (C/G)^2/4 ) ] α = √(RG) [ 1 + ω^2(1/8)( -2(L/R)(C/G) + (L/R)^2 + (C/G)^2 ) ] α = √(RG) [ 1 + ω^2(1/8)((L/R)-(C/G))^2 ] β = ω√(LC) √[{ -RG/LCω^2 + 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 ) }/2] β = ω√(LC) √[{ -RG/LCω^2 + 1 + 1/ω^2 √( ω^4 + (RG/LC)^2 + (ωG/C)^2 + (ωR/L)^2 ) }/2] β = ω√(LC) √[{ -RG/LCω^2 + 1 + (RG/LC)/ω^2 √( ω^4/(RG/LC)^2 + 1 + (ωL/R)^2 + (ωC/G)^2 ) }/2] β = ω√(LC) √[{ -RG/LCω^2 + 1 + (RG/LC)/ω^2 √( 1 + (ωL/R)^2 + (ωC/G)^2 ) }/2] : ωL/R, ωC/G の4次項を削除 β = ω√(LC) √[{ -RG/LCω^2 + 1 + (RG/LC)/ω^2 ( 1 + (ωL/R)^2/2 + (ωC/G)^2/2 ) }/2] : 2項式の1/2乗近似(1との比較) β = ω√(LC) √[{ 1 + (LG/RC)/2 + (CR/LG)/2 }/2] β = ω√(LC) √[ 1/2 + (LG/RC)/4 + (CR/LG)/4 ] 【高周波領域】
α = ω√(LC) √[{ RG/LCω^2 - 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 ) }/2] α = ω√(LC) √[{ RG/LCω^2 - 1 + 1 + (RG/LCω^2)^2/2 + (G/ωC)^2/2 + (R/ωL)^2/2 }/2] :2項式の1/2乗近似(1との比較) α = ω√(LC) √[ RG/LCω^2/2 + (RG/LCω^2)^2/4 + (G/ωC)^2/4 + (R/ωL)^2/4 ] α = ω√(LC) √[ RG/LCω^2/2 + (G/ωC)^2/4 + (R/ωL)^2/4 ]:R/ωLやG/ωCの√4次項を削除 α = ω√(LC) (1/2) √[ 2RG/LCω^2 + (G/ωC)^2 + (R/ωL)^2 ] α = ω√(LC) (1/2) √[ ( G/ωC + R/ωL )^2           ] α = √(LC) (1/2) [ ( G/C + R/L ) ] β = ω√(LC) √[{ -RG/LCω^2 + 1 + √( 1 + (RG/LCω^2)^2 + (G/ωC)^2 + (R/ωL)^2 ) }/2] β = ω√(LC) √[{ -RG/LCω^2 + 1 + 1 + (RG/LCω^2)^2/2 + (G/ωC)^2/2 + (R/ωL)^2/2 }/2] :2項式の1/2乗近似(1との比較) β = ω√(LC) √[ -RG/2LCω^2 + 1 + (RG/LCω^2)^2/4 + (G/ωC)^2/4 + (R/ωL)^2/4 ] β = ω√(LC) √[ 1 - RG/2LCω^2 + (G/ωC)^2/4 + (R/ωL)^2/4 ] :R/ωLやG/ωCの4次項を削除 β = ω√(LC) [ 1 - RG/4LCω^2 + (G/ωC)^2/8 + (R/ωL)^2/8 ] :2項式の1/2乗近似(1との比較) β = ω√(LC) [ 1 + (1/ω^2)( -RG/4LC + (G/C)^2/8 + (R/L)^2/8 ) ]









(*rem0)
---------------------------
:  LC            T(+2)
:  L/C, R/G      T(-6) L(+4) M(+2) I(-4)
:  RC, LG        T(+1)  
:  RG, LCω^2, ωC/G, ωL/R   ---
---------------------------


(*rem1)
√(a+jb) は2乗展開して c+jd と比較すると、
±√(a+jb) = √[{a+√(a^2+b^2)}/2] + j√[{-a+√(a^2+b^2)}/2]