content_image/411820.jpg content_image/411817.jpg content_image/411816.jpg content_image/411812.jpg content_image/411821.jpg content_image/411813.jpg content_image/411811.jpg content_image/411810.jpg content_image/411809.jpg content_image/411808.jpg content_image/411807.jpg content_image/411815.jpg content_image/411819.jpg content_image/411818.jpg content_image/411814.jpg content_image/411806.jpg
We obtain 

and 

<img src='content_image/411820.jpg'>

By summing these equations we obtain: 

<img src='content_image/411817.jpg'>

On the base of definition of the functional Y ′ Fig. 10 and equation (20) we have 

<img src='content_image/411816.jpg'>

As a result we obtain the stationarity property of the functional Y : 

Now we shall discuss the derivative over interaction constant V of functional Y . We shall taken into account the stationarity Y about ˆ Σ and ˆ G c and equation (21). We obtain: 

<img src='content_image/411812.jpg'>

From equation (18) we have: 

and as a consequence we establish 

with the solution 

This constant is F$_{0}$ . Therefore 

with the stationary property 

as in superconducting and in normal states. 

<img src='content_image/411821.jpg'>

<img src='content_image/411813.jpg'>

<img src='content_image/411811.jpg'>

<img src='content_image/411810.jpg'>

<img src='content_image/411809.jpg'>

<img src='content_image/411808.jpg'>

<img src='content_image/411807.jpg'>

As a illustration of our results we shall consider the problem of finding the heat capacity of our strongly cor- related system in normal state and at the low tempera- tures. 

The heat capacity at constant volume V is equal to 

where the entropy S is given by 

The quantity µ is the chemical potential at temperature T . 

At low temperature we may expand F ( µ, V, T ) and µ in even powers of the temperature: 

where µ$_{0}$ is the value of chemical potential for correlated system at T = 0. 

As a result we have 

the linear dependence of the heat capacity of the tem- perature at low its values. Therefore to evaluate the co- efficient γ ( µ$_{0}$, V ) it is necessary to obtain the expansion of F in powers of T by using the expression (31), (19) and Fig.10 for Y ′ obtained by us in previous part of the paper. 

We know that the expression (19) for Y and (31) for thermodynamic potential is stationary with respect to changes in the proper self-energy Σ$_{k}$( iω$_{0}$ ) . Therefore be- cause we are interested in the first corrections to F , we can neglect the explicit temperature dependence of mass operator Σ( k | iω ) and G $^{c}$( k | iω ) and replace them by val- ues Σ( k | iω$_{0}$ ) | T =0 and G $^{c}$( k iω$_{0}$ ) | T =0 calculated at T = 0. Thus the first correction to the T = 0 value of F (31) comes only from the difference between the ω$_{n}$ sums in expression (31) and what we would get if we replace them by integrals according to the equation 

Now it is necessary to consider the functional Y ′ on Fig.10. Since each line of a skeleton diagram of func- tional Y ′ contains an ω$_{n}$ sum, the total first correction to Y ′ is obtained by correcting the computation in each diagram for a single line and use equation (37) for the other ω$_{n}$ sums of the diagram, finally summing over ev- ery line. 

<img src='content_image/411815.jpg'>

## V. HEAT CAPACITY [25] 

<img src='content_image/411819.jpg'>

<img src='content_image/411818.jpg'>

<img src='content_image/411814.jpg'>

<img src='content_image/411806.jpg'>

