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how to prove Log z is piecewise continuous on unit circle mod(z)=1.here z is a complex number

how to prove Log z is piecewise continuous on unit circle mod(z)=1.here z is a complex number

Grade:12

1 Answers

sudhir pal
askIITians Faculty 26 Points
8 years ago
Log z = ln|z| + iArg(z)
since |z| = 1 which implies ln|z| = 0
Log z = iArg(z) Argz = tan-1(y/x) and for x=0 Argz is not defined
for z lying on locus |z| =1 Arg(z) is not defined for z= i & -i hence Log z also not defined
hence Log z is piecewise continuous on unit circle |z| = 1
hence proved
Thanks & Regards
Sudhir,
askIITians Faculty
Qualification.
IIT Delhi

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