aSpaceaSpaceaSpaceaSpace^anSpaceandSpaceandSpaceandSpaceandSpaceandSpaceandSpaceandSpaceandSpaceapplyingSpaceasSpaceasSpaceasSpaceasSpaceauxiliarybeSpacebelowSpaceboundedSpacebySpaceby

^canSpacecompleteSpace^consequenceSpaceConsequentlySpaceConsiderSpacecounterpart

directSpacedomainSpacedomain

everySpaceexpress

firstSpacefollowingcolonSpacefollowingcolonSpacefollowscolonSpaceForSpaceForSpaceforSpaceforSpaceforSpaceforSpaceFUNCTIONS

supset0cdotiven

HARMONICSpacehere

ifSpaceifSpaceInSpaceINCREASINGSpaceintegralSpaceintegralSpaceisSpaceisSpaceisSpaceis

largeperiodSpacelefthyphenhandSpacelefthyphenhandSpaceLEMMASpaceLemmaSpaceLemmaSpaceLemma

makeSpacemapping

zetaprime=2minus1eminuspi/2|eta|0<xi2<xi1leqsigmaeminuspi/2;zetarightarrowlog((zetaminusieta0)minus1)

DeltaRsigmaxieta0etaperiodSpaceRperiodSpacexi*delta*,Space|zeta|zetarightarrow0periodSpace|eta|/xixi1,Spacexi20<xi<xi*|eta0|<deltazeta=xi+ieta|eta|<delta*period=xi+ietanotinDeltazeta=xi+ietanotinDeltaprimezetarightarrow0,SpacezetanotinDeltaprime,Spacezetarightarrow0,SpacezetaSpacesectionDeltasingleendquartationperiod

intu(zetarangleu(xi)fracThetaR(c)dc>0,SpaceDeltaprime={zeta=xi+ieta||eta|<2epi/2xi}periodSpace0<sigma<min(deltaminus|eta0|, log(r01))comma

D=D(eta0, sigma)={zeta|Rezeta>0, |zetaminusieta0|<sigma}periodSpacesectioncelSpace<delta0=min(2delta/3, frac21log(r0minus1))comma

fracpi1logfracxi|eta|+O(1)minus(logfracxi|eta|+O(1))*intu(xiprime)u(xi)fracThetaR(c)dc=fracpi1logfracxi|eta|+O(1)intu(zeta)u(xigammafracThetaR(c)dc+intulanglexiprime)u(xi)fracThetaR(c)dcintu(xi1+ieta0)u(xi.+ieta0)fracThetaD(c)dc|leqfracpi1logfracxi2xi1+frac4piintu(xiprime)u(zeta)fracThetaR(c)dcleq(frac21+o(1))(logfracxi|eta|+O(1))*

needSpaceneed

obtainSpaceofSpaceofSpaceofSpaceofSpaceofSpaceofSpaceOnSpace^onSpaceother

1commaSpace1singleendquartationSpace24periodSpace5Space55Space6Space6periodSpace6periodSpaceleftpar34rightparperiodSpaceleftpar36rightparSpaceleftpar36rightparSpaceleftpar36rightparSpaceleftpar36rightparSpaceleftpar37rightpar

positiveSpaceproofSpaceProof

oatisfiedSpacesatisfiesSpacesecondSpacesetSpacesideSpacesideSpaceSLOWLYSpacesmallSpacesmallSpaceStolzSpaceStolzSpacesuchSpacesuchSpacesufficientlyoufficientlySpacesufficientlySpacesufficiently

CakeSpacetakingSpacethatSpacethatSpaceTheSpaceTheSpacetheSpacetheSpacetheSpacetheSpacetheSpacethenSpaceTheoremSpaceTheoremSpacetheseSpacethis

UNBOUNDED

verifiedperiod

WeSpaceWeSpaceweSpaceweSpace^weSpacewhichSpacewideSpacewillSpacewithSpacewithSpacewithSpacewordscomma