<<peeters` ;
peeters`setGitDir[ "blogit" ]

Manipulate[
Plot[ 
Abs[E^(-I Pi u) + (e + Sqrt[e^2 + 1]) E^(I Pi u)]^2
, {u, -1/2, 1/2}
, PlotRange -> {Full, {0, 100}}
],
{e, 0, 10}]

Clear[rho, t]
rho[u_, e_] = Module[{r, n},
r[uu_, ee_] = Abs[E^(-I Pi uu) + (ee + Sqrt[ee^2 + 1]) E^(I Pi uu)]^2 ;
n[ee_] = Assuming[ ee ≥ 0 && uu ∈Reals, Integrate[ r[uu, ee], {uu, -1/2, 1/2}]] ;
r[u, e]/n[e]
] 

t = Table[ rho[u,e],{e, 0, 10, 2} ] ;

(((Abs[(E^(-I π u))+(e+(Sqrt[1+(e^2)])) (E^(I π u))])^2)/(2 (1+(e^2)+e (Sqrt[1+(e^2)]))))// TraditionalForm

t[[1]] // TraditionalForm
rho[u, e] // TraditionalForm
(*Integrate[t[[1]], {u, -1/2, 1/2}]*)
(*Integrate[t[[3]], {u, -1/2, 1/2}]*)

pp =Plot[ Evaluate@t
, {u, -1/2, 1/2}
,Ticks -> None
, AxesLabel -> {x, None}
, Epilog -> { Text[-Pi/a,{-1/2, -0.5}], Text[-Pi/a,{1/2, -0.5}] }
, PlotRange -> {Full, {-1, 2.5}}
]

peeters`exportForLatex["ps7p2eFig1", pp]