The next step is replacing contour by real time integrals in the
DYSON equation. In that equation one encounters the following contour integrals
3.7.2 LANGRETH Theorem
and their generalizations involving products of three or more terms. To
evaluate (3.53) one can assume that is on the first half of
the contour and is on the latter half. In view of the discussion
of (3.51), we are thus analyzing a
lesser function. The next step is to deform the contour as indicated in
Fig. 3.10. Thus (3.53) becomes
Here, in appending the label to the function B in the first term we made
use of the fact that as long as the integration variable is confined on the
it is less than (in the contour sense) . A similar
argument applies to the second term. Considering the first term in
(3.54) the integration can be split into two parts
where the definition of the retarded function (3.52) has been
used. A similar analysis can be applied to the second term involving contour
, where the advanced function is generated. Putting the two terms
together, one gets the first of LANGRETH's results 
The same result applies for the greater function just by replacing the
labels by the labels. It is easy to generalize the result (3.56)
to a product of three functions.
The retarded and analogously the advanced component of a product of functions defined on the
contour can be derived by repeated use of the definitions
(3.51) and (3.52) and the result (3.56).
In the self-energies another structure occurs
Deformation of contour C
into contours C and C.
where and are contour variables. The derivation of the required
formula is similar to the analysis presented above 
The rules provided by the LANGRETH theorem are summarized in
Rules for analytic continuation derived from the LANGRETH theorem.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors