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WKH# /RUHQW]# WUDQVIRUP# RI# WLPH/ ( ) W′ = γ W − 9[ 2 F5 /# JRHV# RYHU# QRW# WR# WKH# FODVVLFDO# WUDQVIRUP# W′ ≈ W /# DV# LW# VKRXOG# EH/# EXW# WR# D# WUDQVIRUP# RI# WKH# IRUP W′ ≈ W −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′FRV ωW′ − \′VLQ ωW′/ \ = [′VLQ ωW′ + \′FRV ωW′/ ] = ]′/ W = W′ #+:, +WKRXJK/#VWULFWO\#VSHDNLQJ/#LW#LV#QHFHVVDU\#WR#FRQVLGHU#GLIIHUHQWLDO#WUDQVIRUPV#RI#FRRUGLQDWHV#DQG#WLPH,1#,W#LV#FOHDU#WKDW#LQ#WKLV FDVH#WKH#VTXDUHG#GLIIHUHQWLDO#RI#WKH#LQWHUYDO#LV#QRW#UHWDLQHG#+WKH#FRHIILFLHQW#RI# F5GW′5 #LV#QRW#HTXDO#WR#XQLW\,= ( ) GV5 = 4 −  ω5 F5 [′5 + \′5   F5 GW ′5 − 5ω ( [ ′G\ ′ − \′G[′)GW′ − G[ ′5 − G\ ′5 − G] ′5  ( ) =  4  + 5 ω F5   \′ G[′ GW ′ − [′ G\′ GW ′   − ω5 F5 [′5 + \′5   F 5 GW ′5 − G[′5 − G\ ′5 − G] ′5  #+;, ≠ GV′5 = F5GW′5 − G[′5 − G\′5 − G]′5 1 7KHUHIRUH/# WKLV# WUDQVIRUPDWLRQ# LV# D# PHUH# PLVXQGHUVWDQGLQJ1# :H/# IRU# H[DPSOH/# LQ# FDOFXODWLQJ# WKH# FRPSRQHQWV# RI# WKH PHWULF#WHQVRU#RI#D#PRYLQJ#IUDPH#RI#UHIHUHQFH/#XVH#QRW#WKH#FODVVLFDO#WUDQVIRUPV G[ = G[′ − 9GW′/ G\ = G\′/ G] = G]′/ GW = GW′ /# #+<, EXW#WKH#UHODWLYLVWLF#RQHV G[ = G[′ − 9GW′ / G\ = G\′/ G] = G]′/ GW = GW′ − 9G[′ 2 F5 /# #+43, 4−9 5 2 F5 4−9 5 2 F5 DV# ZH# UHDOL]H# WKDW# LQ# WKH# ILUVW# FDVH# WKH# VTXDUHG# GLIIHUHQWLDO# RI# WKH# LQWHUYDO# ZRXOG# EH# H[SUHVVHG# LQ# WKH# VDPH# ZD\# DV/# IRU H[DPSOH/#LQ#>9@#RU#>:@/#WKDW#LV/#ZLWK#DQ#LPSURSHU#PHWULF/#DQG/#WKXV/#LWV#ZKROHQHVV#DQG#LQYDULDQFH#ZRXOG#EH#EURNHQ= GV5 = 4  − 95 F5   F5 GW ′5  + 59G[′GW′ − G[′5 − G\′5 − G]′5 #+44, =  4  + 5 9 F5 G[′ GW′ − 95 F5   F5 GW ′5  − G[′5 − G\′5 − G]′5 ≠ GV ′5 = F 5 GW ′5 − G[′5 − G\′5 − G]′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−G[5 − G\5 − G] 5 =  − −5  ω F5   \ ′ G[′ GW′ − [′ G\ ′ GW′   + ω5 F5 [′5 + \′5   F 5 GW ′5 − G[ ′5 − G\ ′5 − G] ′5 /#  #+45, DQG#WKHQ#VROYH#WKLV#HTXDWLRQ#LQ#FRPELQDWLRQ#ZLWK#WKH#HTXDWLRQ F5GW5 − G[5 − G\5 − G]5 = F5GW′5 − G[′5 − G\′5 − G]′5 # 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