Logarithmic differential equation

[muppet]

Hi all,
I have functions $\eta(\mu),Z(\mu)$ related by
$$\eta(\mu)=-\frac{d \ln{Z}}{d \ln{\mu}}$$
I'm told that if we specify $\eta$ then we have
$$Z^{-1}(\mu)=Z^{-1}(\mu_0)\exp(\int^{\mu}_{\mu_0} dk \ \eta(k))$$
but upon inverting this equation, taking the log and differentiating wrt $\ln(\mu)$ I get
$$-\frac{d \ln{Z}}{d \ln{\mu}}=-\mu \frac{d }{d \mu}(-\int^{\mu}_{\mu_0} dk \ \eta(k))=\mu \eta(\mu)$$
What am I doing wrong?
Thanks in advance.

 

[I like Serena]

Hi muppet!

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