Consider a function so constructed such that,
\(Z_{s}=\cfrac { d\, \beta e^{ f(E_{ s }) } }{ d\, E_{ s } } =\beta f^{ ' }(E_{ s })e^{ f(E_{ s }) }\)
where \(\beta\) is a constant, and \(f(E_{s})\) is a function in \(E_s\) with roots \(E_{si}\) \(i=1,2,3...\). ie.
\( f(E_{ s }=E_{ { si } })=0\quad \quad \quad i=1,2,3...\)
then
\( e^{ f(E_{ s }=E_{ { si } }) }=e^{ 0 }=1\)
and,
\( \left[{ \cfrac { d\, \beta e^{ f(E_{ s }) } }{ d\, E_{ s } } }\right]_{ E_{ si } }=\beta f^{ ' }(E_{ s }=E_{ { si } }).1=Z_{ si }\)
Since differentiation by product rule leaves behind one term without each root factor, we define,
\( f^{ ' }(E_{ si })=p(Z_{ si }).(E_{ si }-E_{ s1 })(E_{ si }-E_{ s2 })..\)
except for the factor of \((E_{si}-E_{si})\) such that,
\( f^{ ' }(E_{ si })=p(Z_{ si }).\prod \limits_{\substack{ j=1 \\ j\ne i }}^n (E_{ si }-E_{ { sj} })=\cfrac{Z_{ si }}{\beta}\)
where \(p(Z_{si})\) is defined by,
\( p(Z_{ si })=\cfrac { Z_{ si } }{ \beta .\prod\limits _{\substack{ j=1 \\ j\ne i }}^n (E_{ si }-E_{ { sj} }) } \)
where we insist that \(E_{si}\ne E_{sj}\) for \(i\ne j\), each root of \(f(E_s)\) is unique. \(p(Z_{si})\) can also be defined as \(p(E_{si})\) as
\(p'(E_{s})\prod_i(E_s-E_{si})\) is zero for all \(E_s=E_{si}\). We will still have,
\( p(E_{ si })=\cfrac { Z_{ si } }{ \beta .\prod\limits _{\substack{ j=1 \\ j\ne i }}^n (E_{ si }-E_{ { sj} }) } \)
\(\beta\) is defined such that,
\( \int _{ 0 }^{ E_s max }{ Z_{ s } } d\, E_{ s }=Z=\left[ \beta e^{ f(E_{ s }) } \right] ^{ E_s max }_{ 0 }=\sum\limits _{ i }^{ all\, i } Z_{ si }\)
\(\beta=\cfrac{Z}{\left[ e^{ f(E_{ s }) } \right] ^{ E_s max }_{ 0 }}\)
If \(e^{ f(E_{ s }max)}\rightarrow0\) for \(E_{ s }max\rightarrow \infty\)
\(\beta=\cfrac{Z}{0-e^{ f(0) }}=-\cfrac{Z}{e^{ f(0) }}\)
at \(e^{f(E_s=0)}\).
Notice the expression for \(Z\) is always zero integrating from root to root of \(f(E_s)\). This is because \(e^{ f(E_{ s })}\) intersect \(y=1\) repeatedly such that the area above and below the x-axis of \(e^{ f(E_{ s })}\) is equal, from root to root of \(f(E_s)\). This means, the value of \(Z\) is determined by that part of the curve from zero to the first root of \(f(E_s)\) and, the last root of \(f(E_s)\) to the tail end of \(e^{ f(E_{ s })}\) towards infinity.
If the first root is zero then \(\beta\) can be found from,
\(\beta=\cfrac{Z}{0-e^{ f(E_s\,last) }}=-\cfrac{Z}{e^{ f(E_s\,last) }}=-Z\)
because we know that the integration of \(Z_s\) with boundaries between roots of \(f(E_s)\) sums to zero.
No need to involve \(T\), which ever way temperature is defined.
