Integrate[ Exp[I a u ] Sqrt[1-u^2],{u,-1,1}]

                      Pi BesselJ[1, a]
Out[1]= If[Im[a] > 0, ----------------,
                             a

                I a u           2
>    Integrate[E      Sqrt[1 - u ], {u, -1, 1}, Assumptions -> Im[a] <= 0]]

Simplify[%,Element[a,Reals]]

        Pi BesselJ[1, a]
Out[2]= ----------------
               a

Plot[%^2,{a,0,10},PlotRange->All]
Plot[{20 %%^2+1,1},{a,4,10},PlotRange->All]
Show[%%,%]

N[(Pi/2)^2]
Out[6]= 2.4674

FindRoot[BesselJ[1, a]==0,{a,4}]

Out[7]= {a -> 3.83171}

a/Pi /. %

Out[8]= 1.21967

In[9]:= Export["circular_aperture.eps",%5]

Out[9]= circular_aperture.eps
