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so .. if f (z) = u + iv is analytic on D, then u and v are harmonic on D...

now ...

if f (z) never vanishes on the domain ...

then show log |f (z)| is harmonic on the domain ...

Recall: harmonic means second partial derivative of f with respect to x + second partial derivative of f with respect to y = 0

umm? did they mean to say that harmonic means second partial derivative of log |f (z)| with respect to x + second partial derivative of log |f (z)| with respect to y = 0

now ...

if f (z) never vanishes on the domain ...

then show log |f (z)| is harmonic on the domain ...

Recall: harmonic means second partial derivative of f with respect to x + second partial derivative of f with respect to y = 0

umm? did they mean to say that harmonic means second partial derivative of log |f (z)| with respect to x + second partial derivative of log |f (z)| with respect to y = 0

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