$N=\displaystyle\sum_r\displaystyle\frac{1}{e^{\alpha+\beta\epsilon_r}+1}$

N=sum_r 1/(e^(alpha+beta*e_r)+1)

$ n_r =\displaystyle\frac{1}{e^{ \alpha + \beta \epsilon_r }+1}$

n_r =1/(exp( alpha + beta * epsilon_r )+1)

$ \vec{p} = \hbar \vec{k} $

&p = hbar * &k

$ \epsilon =\displaystyle\frac{ \hbar ^2 \vec{k} ^2}{2 m }$

e = hbar ^2 &k ^2/(2* m )

$d^3 n =2\displaystyle\frac{ V }{(2 \pi )^3}d^3 k $

d^3 n =2* V /((2* pi )^3)*d^3 k

$ N =2\displaystyle\frac{ V }{(2 \pi )^3}\displaystyle\frac{4 \pi }{3} k_F ^3$

N =2* V * 4 * pi * k_F ^3/(3*(2 \pi )^3)

$ k_F =\left(3 \pi ^2\displaystyle\frac{ N }{ V }\right)^{1/3}$

k_F =(3* pi ^2 * N / V )^(1/3)

$ \lambda_F =\displaystyle\displaystyle\frac{2 \pi }{ k_F }$

lambda_F = 2* pi / k_F

$ \epsilon_F =\displaystyle\frac{ \hbar ^2}{2 m }\left(3 \pi ^2\displaystyle\frac{ N }{ V }\right)^{2/3}$

epsilon_F = hbar ^2/(2* m )*(3* pi ^2* N / V )^(2/3)

$ T_F =\displaystyle\frac{ \epsilon_F }{ k_B }$

T_F = epsilon_F / k_B

$ \displaystyle\sum_ i \rightarrow\displaystyle\frac{ 2 V }{4 \pi ^2}\left(\displaystyle\frac{2 m }{ \hbar ^2}\right)^{3/2} \epsilon ^{1/2} d\epsilon $

$ \ln Z_{FD} = \alpha N + \displaystyle\frac{ V }{2 \pi ^2}\left(\displaystyle\frac{2 m }{ \hbar ^2}\right)^{3/2}\displaystyle\int_0^{\infty}\ln(1+e^{- \beta \epsilon - \alpha }) \epsilon ^{1/2} d\epsilon $

ln Z_FD = alpha * N + ( V /2* pi ^2)*(2* m / hbar ^2)^(3/2)*@INTEGARTE(ln(1+exp(- beta * epsilon - alpha ))* epsilon ^(1/2), epsilon , 0 , infty )

$ N =\displaystyle\frac{ V }{2 \pi ^2}\left(\displaystyle\frac{2 m }{ \hbar ^2}\right)^{3/2}\int_0^{\infty}\displaystyle\frac{ \epsilon ^{1/2} d\epsilon }{e^{ \beta \epsilon + \alpha }+1}$

N =\displaystyle\frac{ V }{2 \pi ^2}\left(\displaystyle\frac{2 m }{ \hbar ^2}\right)^{3/2}\int_0^{\infty}\displaystyle\frac{ \epsilon ^{1/2} d\epsilon }{e^{ \beta \epsilon + \alpha }+1}