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I have functions [itex]\eta(\mu),Z(\mu)[/itex] related by

[tex]\eta(\mu)=-\frac{d \ln{Z}}{d \ln{\mu}}[/tex]

I'm told that if we specify [itex]\eta[/itex] then we have

[tex]Z^{-1}(\mu)=Z^{-1}(\mu_0)\exp(\int^{\mu}_{\mu_0} dk \ \eta(k))[/tex]

but upon inverting this equation, taking the log and differentiating wrt [itex]\ln(\mu)[/itex] I get

[tex]-\frac{d \ln{Z}}{d \ln{\mu}}=-\mu \frac{d }{d \mu}(-\int^{\mu}_{\mu_0} dk \ \eta(k))=\mu \eta(\mu)[/tex]

What am I doing wrong?

Thanks in advance.

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# Logarithmic differential equation

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