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(hatxi,hateta)inC2rho=minusdelta(2minusdelta).hathinC2(overlineOmegar),Space(hatxi,hateta)inC2,Space(hatxi,hateta)inC2periodSpaceLsigma(p;(xi, eta))fracpartialwpartialoverlinewpartial2rho=1;zeta=exp(iln|z|2)delta=0,Spacep=(z, 0)inMrperiodSpacefracpartialzkpartial2rho=2frac|z|2deltasemicolon

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zinC,SpacedeltainRfracpartialzpartialrho=0;(hatxi, deltahateta)(xi, eta)inC2

p=(z, minusdeltaexp(iln|z|2))]periodSpacefracpartialwpartialrho=(1minusdelta)exp(iln|z|2);fracpartialzpartialoverlinewpartial2rho=fracZiexp(iln|z|2)semicolon

[)=(z, w)=(z, minusdeltaexp(iln|z|2))

+taudeltahath+tau(1minustau)frac2minusdelta(1minusdelta)2hath)|eta|2}.hatsigma(z, w)=minustildeh(|z|2, w)(minusrhor(z, w))1/iota

+(2minusdelta)(minusdelta2(2minusdelta)fracpartialwpartialoverlinewpartial2hath+2taudelta(1minusdelta)Re(zetafracpartialwpartialhath)

+2delta(2minusdelta)Re[(minusdelta(2minusdelta)fracpartialzpartialoverlinewpartial2hath+tau(1minusdelta)fracpartialzpartialhath+taufracZizetahath)xinotin

gsigma{p;(xi, eta))=deltatauminus2(2minusdelta)tauminus2(delta2(2minusdelta)(minus(2minusdelta)fracpartialzpartialoverlinezpartialhath+2tauhathfrac|z|21)|xi|2

+(minusdelta2(2minusdelta)fracpartialwpartialoverlinewpartial2hath+2taudelta(1minusdelta)Re(zetafracpartialwpartialhath)+taudeltahath+tau(1minustau)frac2minusdelta(1minusdelta)2hath)|hateta|2geqq0

(minus2fracpartialzpartialoverlinezpartial2hath+2frac|z|2tauhath)|xi|2+2Re([tauzeta(fracpartialzpartialhath+fracZihath)hatxihateta]+frac21tau(1minustau)hath|hateta|2geqq0

(minus(2minusdelta)fracpartialz5overlinezpartial2hath+2frac|z|2tauhath)|xi|2+2Re([(minusdelta(2minusdelta)fracpartialwpartialoverlinewpartial2hath+tau(1minusdelta)fracpartialzpartialhath+taufracZizetahath)hatxihateta

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