Contours are used when you need to evaluate some definite integral along the real axis.

(I dont know what your final interest is).

However the proceedure is to use a closed contour using the theory of Residues, from complex analysis.

In that case you must show that the rest of the closed contour integral gives contribution zero,

as the magnitude of z is extended to infinity, in the upper half plane. You go around the region of singularity,

in this case seems z=0. Other singularities from the sine function.

Usually a vertical contour cutting the real z-axis between 0 and 1 is a vertical straight line in the complex plane parallel to the imaginary axis whose intersection with the real axis is between 0 and 1.

In other words it is a line with parametrization t -> c+it, with 0

Many thanks, both professors, for your answers. I suppose that we need to integrate firstly under the contour L based on the line before integrating another variable under the limits 0 and 00(infinite). I am working wigb the Fermi-Dirac integral,

It seems like you're describing a vertical contour "L" in the complex plane that intersects the real axis between 0 and 1. In complex analysis, contours are paths in the complex plane that are often used for integration and studying the behavior of complex functions.

If "L" is a vertical line segment that starts at a point on the real axis at 0 and ends at a point on the real axis at 1, you could parameterize it as a function of a real variable "t" in the following way:

L(t) = t + i * y,

where "y" is the vertical coordinate of the complex number, which remains constant along the line. Similar to what is proposed by Giovanni Dore

This contour "L" can be useful when performing integrals of complex functions that are defined along the real axis, particularly between 0 and 1. It allows you to consider integrals over this specific path and analyze the behavior of the function within that interval.