Spherical Bessel Functions j, n, h, h*

Plot[Evaluate[Table[BesselJ[n, x], {n, 4}]], {x, 0, 10}];

j[l_, x_]  :=  Sqrt[Pi/(2 x)] BesselJ[l + 1/2, x]

j[0, x]

\(\(	j[1, x]\)\)

\(\(	j[2, x]\)\)

Integrate[ x^2  j[1, x]^2,  {x, 0, Infinity}]

\(\(Plot[Evaluate[Table[
			\ j[l, x], {l, 0, 4}
		]], {x, 0, 10}, 
      \ PlotRange\  -> \ {\(-0.3\), 1}]; \)\)

\(\(	n[l_, x_]\  := 
      \((\(-1\))\)^\((l + 1)\)\ \ Sqrt[Pi/\((2\ x)\)]\ 
        BesselJ[\(-l\) - 1/2, x]\)\)

\(\(	n1 = \ n[0, x]\)\)

\(\(	n2 = n[1, x]\)\)

\(\(Plot[Evaluate[Table[
			\ n[l, x], {l, 0, 4}
		]], {x, 0, 10}, 
      \ PlotRange\  -> \ {\(-1\), 0.4}]; \)\)

h[l_, x_]  :=  Simplify[TrigToExp[  j[l, x]  +  I n[l, x] ]]

hstar[l_, x_]  :=  
      Simplify[TrigToExp[  j[l, x]  -  I n[l, x] ]]

h0  = h[0, x]

h1  =  h[1, x]

h[0,  I k r]

hstar[0, I k r ]

Information["Plot", LongForm -> True]

Integrate[ 1/(2 Pi) E^(I x Cos[t]),  {t, 0, 2  Pi}]

Integrate[ 1/(2 Pi) E^(I x  Cos[t]) Cos[ t], 
       {t, 0, 2 Pi}]

Integrate[ 1/(2 Pi) E^(I x  Cos[t]) Cos[2 t], 
       {t, 0, 2 Pi}]

Integrate[ 1/(2 Pi) E^(I x  Cos[t]) Cos[3 t], 
       {t, 0, 2 Pi}]