f1[t_] := t;
f2[t_] := t^2;
f3[t_] := t^3;
f4[t_] := UnitStep[t + Pi/2] - UnitStep[t - Pi/2];
f5[t_] := UnitStep[t] - UnitStep[t - Pi/2];
f6[t_] := UnitStep[t] - UnitStep[t - Pi/4];
f6[t_] := UnitStep[t] - UnitStep[t - Pi/8];
f7[t_] := UnitStep[t] - UnitStep[t - Pi/16];
f8[t_] := Sin[t];
f9[t_] := Cos[t];
f10[t_] :=  Sin[t] Cos[t];
f11[t_] :=  
        Sin[t] Cos[t]^2  + Exp[t];
f12[t_] :=  Exp[-Abs[t]];
f13[t_] :=  .5^Abs[t];
f14[t_] := 5;
f15[t_] := DiracDelta[t]

ticks = {-4 Pi, -π, -(π/2), 
          0, (π/2), π, 4  Pi};
\(\(fticks = {\(-4\)\ Pi, \ \(-π\), \(-\(π\/2\)\), 
          0, π\/2, π, \ 4  Pi};\)
\)
\(\(F[u_, T_, from_, 
          to_] := \ \(1\/T\) \(∫\_from\%to g[
                t] \(E\^\(\(-2\) I\ π\ t\ u\/T\)\) \[DifferentialD]t\);\)
\)
\(do[f_, T_, from_, to_] := 
      Module[{}, 
g = f; 
freq = 
          Plot[Evaluate[F[u, T, from, to]], {u, \(-4\) Pi, 
              4  Pi}, 
DisplayFunction -> 
              Identity, 
Ticks -> {fticks, 
                Automatic}, 
PlotRange -> 
              All, 
PlotLabel -> "|F(ω)|"]\ ; 

p1 = 
          Plot[Evaluate[f[t]], {t, \(-4\) Pi, 
              4  Pi}, 
PlotRange -> 
              All, 
PlotLabel -> "f(t)", 
Ticks -> {ticks, 
                Automatic}, 
DisplayFunction -> 
              Identity]; 

Show[
          GraphicsArray[{p1, freq}], 
          DisplayFunction -> $DisplayFunction]
]\)

from = -4 Pi;  to = 4 Pi; T = 8 Pi;
do[f1, T, from, to];
do[f2, T, from, to];
do[f3, T, from, to];
do[f4, T, from, to];
do[f5, T, from, to];
do[f6, T, from, to];
do[f7, T, from, to];
do[f8, T, from, to];
do[f9, T, from, to];
do[f10, T, from, 
        to];

do[f15, 
      4  Pi, -∞, ∞]

2/(1 + (2 Pi (1/2 Pi))^2) // N

?PlotStyle

?Plot

Transpose[{{1, 2, 3}, {4, 5, 6}}]

{{1, 2, 3}, {4, 5, 6}} // MatrixForm

Transpose {x, N[(((Abs[freq]))^2)]}

? Transpose

?Version

System`Version

System`Version

$Version

?ParametricPlot

?Table

Transpose[Table[i, {i, -Pi, Pi, Pi/8}], mag]

Length[Table[i, {i, -Pi, Pi, Pi/8}]]
Length[mag]

Transpose[{1, 2}, {2, 3}]

Needs["Graphics`Graphics`"]
\(Clear[pairs, x, y]
\)
y = {1.0, 0.0, 1.0, 2.0, 1.0}
x = {0.0, 1.0, 1.0, 1.0, 2.0}
\(pairs = 
      Table[{x[\([i]\)], y[\([i]\)]}, {i, 1, 4}]
\)
LabeledListPlot[pairs]

freq=Table[
        F[u, f2], {u, -Pi, π, π/8}]

N[%]

?Xticks

?Ticks

?Abs

r = 
      FullSimplify[(∫_(-∞)^∞) 
            A  (E^(((-(u^2))/(2  (σ^2))))) (E^(-I  2 Pi x u))  \[DifferentialD]u, 
        Element[{u, x, σ, M}, Integers]]

Limit[%, {M, ∞}]

Erf[(∞/Sign)[σ]] // N

(∫_(-∞)^∞) 
      A (E^((-u^2/(2 a^2)))) (E^(I 2 π u x)) \[DifferentialD]u

(E^((-u^2/(2 a^2)))) (E^(I 2 π u x))

FullSimplify[(1/(4 (π^2))) (∫_(-∞)^∞) (E^(-((u^2)/(2 (a^2))) + 
                  2 I π u x)) \[DifferentialD]u, 
      Element[a > 0]]

FullSimplify[(∫_(-∞)^∞) 
            Exp[I 2 π (u x + 
                  v y)] \[DifferentialD]u, 
      Element[{u, v}, Reals]]

Limit[%, M -> ∞]

?DiracDelta*

Plot[Exp[Abs[x]], {x, -3, 3}]

?Gauss*

σ = 6;  α =  .4;  A = 7;  
    B = 1;
Plot[ 
      B  (E^(((-(t^2))/(2  (σ^2))))) - 
        A (E^(((-(t^2))/(2  (α^2))))) , {t, -3, 
        3}]