Mathematica

Full Simplify

Alpha12 = 1/\[Sqrt]2  (\[Alpha]1 - I* Exp[I \[Phi]] \[Alpha]2)*Exp[I \[CapitalDelta]];

Alpha12C = Conjugate[Alpha12];

Assuming[{\[Alpha]1 && \[Alpha]2 && \[Phi] && \[CapitalDelta]} > 0, FullSimplify[Alpha12C]]

FullSimplify[Alpha12C, Assumptions -> {\[Phi], \[Alpha]1, \[Alpha]2, \[CapitalDelta]} \[Element] Reals]

Integration

P = (1/(\[Sqrt](2*\[Pi])*dr)) Exp[-(x - x0)^2/(2*dr^2)]

Assuming[{x0 && dr} > 0, Integrate[P, {x, -\[Infinity], \[Infinity]}]]

Coupled differential equation

sol = DSolve[{x'[t] == a*y[t], y'[t] == b*z[t] - c, z'[t] == -b*y[t],  x[0] == 0, y[0] == 0, z[0] == 0}, {x[t], y[t], z[t]}, t]

Simplify[sol]

FOR Loop

P = 2/(a*b) Exp[-x^2/a] Exp[-y^2/b]

g = {};

For[x = -10; a = 2; b = 4, x <= 10, x += 1, P; AppendTo[g, P]];

h = {};

For[y = -10, y <= 10, y += 1, g; AppendTo[h, g]];

h;

ListPlot3D[h, PlotRange -> All]

Integrating one of the variables and plot as a function of other variable

A[\[Theta]_, \[Phi]_] := Cos[\[Theta] + \[Phi]];

B[\[Phi]_] := Integrate[A[\[Theta], \[Phi]], {\[Theta], 0, \[Pi]}];

Plot[B[\[Phi]], {\[Phi], 0, 2 \[Pi]}]