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Logarithmic differential equation

  1. Nov 2, 2011 #1
    Hi all,
    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.
  2. jcsd
  3. Nov 2, 2011 #2

    I like Serena

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    Homework Helper

    Hi muppet! :smile:

    Your integration is off.
    It should be:
    [tex]Z(\mu)^{-1}=Z(\mu_0)^{-1} \cdot {1 \over \mu} \cdot \exp(\int^{\mu}_{\mu_0} dk \ \eta(k))[/tex]
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