Statistical mechanics. Partition function.

[LagrangeEuler]

Homework Statement

If ##Z## is homogeneous function with property
##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##
and you calculate Z(T,V,N). Could you calculate directly ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##.

Homework Equations

##Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k T)^{\frac{3N}{2}}\int...\int_{V} e^{-\frac{U}{kT}}##

The Attempt at a Solution

When I have for exaple
##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)=\frac{1}{h^{3N}N!}(2\pi m k \alpha T)^{\frac{3N}{2}}\int...\int_{\alpha^{-\frac{3}{\nu}}V} e^{-\frac{U}{k\alpha T}}##
I'm not sure how to integrate this ##e^{-\frac{U}{k\alpha T}}## in exponent.
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[LagrangeEuler]

From this if I understand well
##\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}##
need to be equal
##\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}=\alpha^{-\frac{3N}{\nu}}...##