High Brightness Electron Beam Diagnostics and their ... - CASA

Unfortunatelylog[^S(!)]cannotbeexpressedasaCauchyintegralbecauselog[^S(!)]isnotan dispersionrelationshavebeenappliedtothisfunctiontoretrieveitsimaginarypart(i.e.thephase of^S(!)). analyticinthefullupperhalf-plane.Moreover: Obviouslylog[^SS()]isnota\good"function!Acommonlyusedruseistodenethefunction Ilog[~S()] ()def jj!1 =log^S()log^S() !Z0log[~S()]6=0 (C.5)
whereisanarbitrarypointoftheupperhalf-planewherelog[~S]isanalytic(notethat()isnot Thedispersionrelationsappliedto()yields: singularat=). i()=PZ+1 1(x)dx x (C.7) (C.6)
UsingEqn.(C.6)weget: whichexpandsto: ilog[^S()]=ilog[^S()]+()24PZ+1 ilog[^S()]=ilog[^S()]+()24PZ+1 1log[^S(x)]dx 1log[^S(x)]dx (x)(x)PZ+1 (x)(x)log[^S()]PZ+1 1log[^S()]dx (x)(x)35(C.8)
ThetwofarRHStermsarezero,sothatwenallyget: log[^S()]=log[^S()]+iPZ+1+log[^S()]PZ+1
1dx 1dx x(C.9) x
relatelog[j^S()j]tothephaseof^S(),(): Byidentifyingtherealandimaginarypartsweobtainthedispersionrelationsforlog[^S()]which log[j^S()j]=log[j^S()j]+PZ+1 1log[^S(x)]dx 1 (x)(x) (x)(x) (x)dx (C.10)
whichaftertakingintoaccountthesymmetryof^S(^S()=^S()yieldsfor=0: ()=()1()PZ+1 ()=(0)2PZ1 1logj^S(x)jlogj^S()j 0logj^S(x)jdx (x)(x)dx x22 (C.12)
(C.11)