Logarithmic differential equation

  • Thread starter muppet
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  • #1
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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.
 

Answers and Replies

  • #2
I like Serena
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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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