(*^
::[	Information =

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	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

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Setup:
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Needs["Calculus`VectorAnalysis`"]
Needs["Calculus`DSolve`"]
Needs["Algebra`Trigonometry`"]
Needs["Graphics`Graphics`"]
Needs["Graphics`Graphics3D`"]
Needs["Graphics`ParametricPlot3D`"]
:[font = section; inactive; preserveAspect; startGroup]
Example 1a: Cartesian
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect; startGroup]
SetCoordinates[Cartesian]
:[font = output; output; inactive; preserveAspect; endGroup]
Cartesian[x, y, z]
;[o]
Cartesian[x, y, z]
:[font = input; preserveAspect]
psi[x,y,z] = X[x] Y[y] Z[z];
:[font = input; preserveAspect; startGroup]
eq1= Laplacian[ psi[x,y,z] ]/psi[x,y,z] //Expand
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][X][x]/X[x] + Derivative[2][Y][y]/Y[y] + Derivative[2][Z][z]/Z[z]
;[o]
X''[x]   Y''[y]   Z''[z]
------ + ------ + ------
 X[x]     Y[y]     Z[z]
:[font = input; preserveAspect]
select[eq_,var_]:= Select[eq,(!FreeQ[#,var])&]
:[font = input; preserveAspect; startGroup]
xEq= select[eq1,x] == -cx^2
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][X][x]/X[x] == -cx^2
;[o]
X''[x]       2
------ == -cx
 X[x]
:[font = input; preserveAspect; startGroup]
yEq= select[eq1,y] == -cy^2
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][Y][y]/Y[y] == -cy^2
;[o]
Y''[y]       2
------ == -cy
 Y[y]
:[font = input; preserveAspect; startGroup]
zEq= select[eq1,z] == +cz^2
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][Z][z]/Z[z] == cz^2
;[o]
Z''[z]      2
------ == cz
 Z[z]
:[font = subsection; inactive; preserveAspect]
Note: there is a constraint:  -cx^2 -cy^2 +cz^2 ==0
:[font = input; preserveAspect; startGroup]
xEq2= Thread[xEq X[x],Equal]
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][X][x] == -(cx^2*X[x])
;[o]
              2
X''[x] == -(cx  X[x])
:[font = input; preserveAspect; startGroup]
xSol= DSolve[xEq2,X[x],x][[1]]
:[font = output; output; inactive; preserveAspect; endGroup]
{X[x] -> E^(-I*cx*x)*C[1] + E^(I*cx*x)*C[2]}
;[o]
          -I cx x         I cx x
{X[x] -> E        C[1] + E       C[2]}
:[font = input; preserveAspect; startGroup]
xSol //ComplexToTrig //Simplify
:[font = output; output; inactive; preserveAspect; endGroup]
{X[x] -> C[1]*Cos[cx*x] + C[2]*Cos[cx*x] - I*C[1]*Sin[cx*x] + I*C[2]*Sin[cx*x]}
;[o]
{X[x] -> C[1] Cos[cx x] + C[2] Cos[cx x] - I C[1] Sin[cx x] + I C[2] Sin[cx x]}
:[font = input; preserveAspect; startGroup]
zEq2= Thread[zEq Z[z],Equal]
:[font = output; output; inactive; preserveAspect; endGroup]
Derivative[2][Z][z] == cz^2*Z[z]
;[o]
            2
Z''[z] == cz  Z[z]
:[font = input; preserveAspect; startGroup]
zSol= DSolve[zEq2,Z[z],z][[1]]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{Z[z] -> C[1]/E^(cz*z) + E^(cz*z)*C[2]}
;[o]
         C[1]     cz z
{Z[z] -> ----- + E     C[2]}
          cz z
         E
:[font = section; inactive; preserveAspect; startGroup]
Example 1b: Cylindrical
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect; startGroup]
SetCoordinates[Cylindrical]
:[font = output; output; inactive; preserveAspect; endGroup]
Cylindrical[r, theta, z]
;[o]
Cylindrical[r, theta, z]
:[font = input; preserveAspect; endGroup]
psi[r,theta,z] =R[r] Theta[theta] Z[z];
:[font = section; inactive; preserveAspect; startGroup]
Example 1c: Spherical
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect; startGroup]
SetCoordinates[Spherical]
:[font = output; output; inactive; preserveAspect; endGroup]
Spherical[r, theta, phi]
;[o]
Spherical[r, theta, phi]
:[font = input; preserveAspect]
psi[r,theta,phi] =R[r] ang[theta,phi];
:[font = subsection; inactive; preserveAspect]
We will show the solution of this equation is SphericalHarmonicY
:[font = input; preserveAspect]
rule= {ang -> (SphericalHarmonicY[el,m,#1, #2]&)}
:[font = input; preserveAspect]
eq4= eq3 /.rule //Simplify
:[font = subsection; inactive; preserveAspect]
We do this for a range of {el,m} and assume it generalizes.
:[font = input; preserveAspect; endGroup]
Table[ eq4 ,{el,1,6},{m,0,el-1}] //Simplify //N //Chop
:[font = section; inactive; preserveAspect; startGroup]
Example 4:  SphericalHarmonicY
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
plot[l_,m_]:=
SphericalPlot3D[ 
       SphericalHarmonicY[l,m,theta,phi] //Abs //Evaluate
      ,{theta,0,Pi}
      ,{phi,0,2 Pi}
      ,Boxed->False
      ,Axes->False
      ];
:[font = input; preserveAspect; startGroup]
plot[5,1];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 119; pictureHeight = 174; endGroup]
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Print["{l,m}=",{l,m}];
plot[l,m];

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Problem 5: SphericalHarmonicY
(I've done the example here with Sin. You need to change it.
This serves as a templave for many of the other problems.)
;[s]
2:0,0;30,1;150,-1;
2:1,19,14,Times,1,18,0,0,0;1,13,9,Times,3,12,0,0,0;
:[font = input; preserveAspect]
Clear["Global`*"];
:[font = text; inactive; preserveAspect]
Note: the conjugate of  SphericalHarmonicY[2,m, theta, phi]   
= (-1)^m SphericalHarmonicY[2,-m, theta, phi]
:[font = input; preserveAspect]
doit[m_,n_]:= 
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Table[ doit[n,m] ,{n,0,3},{m,0,3}] //TableForm
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TableForm[{{0, 0, 0, 0}, {0, Pi, 0, 0}, {0, 0, Pi, 0}, {0, 0, 0, Pi}}]
;[o]
0   0    0    0

0   Pi   0    0

0   0    Pi   0

0   0    0    Pi
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Plot[ Table[ Sin[n theta] ,{n,0,3,1}] //Evaluate 
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Mfstroke
P
P
P
% End of Graphics
MathPictureEnd
:[font = section; inactive; preserveAspect; startGroup]
Problem 6: Rayleigh's Formula
Choose k=1, and pick a range for the integration that looks nice. 
Take about 15 terms in the seties
;[s]
2:0,0;30,1;131,-1;
2:1,19,14,Times,1,18,0,0,0;1,13,9,Times,3,12,0,0,0;
:[font = input; preserveAspect; endGroup]
Clear["Global`*"];
:[font = section; inactive; preserveAspect; startGroup]
Problem 7: LegendreP
:[font = input; preserveAspect; endGroup]
Clear["Global`*"];
:[font = section; inactive; preserveAspect; startGroup]
Problem 8: Expansion in LegendreP
z=Cos[theta]
;[s]
3:0,0;34,1;46,0;47,-1;
2:2,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = input; preserveAspect; endGroup]
Clear["Global`*"];
:[font = section; inactive; preserveAspect]
Problem 9: Expansion in LegendreP
z=Cos[theta]
;[s]
2:0,0;34,1;47,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = section; inactive; preserveAspect; startGroup]
Problem 10:
:[font = input; preserveAspect]
Clear["Global`*"];
:[font = input; preserveAspect; endGroup]
(* This is the preferred weighting to make "nice" plots *)
Do[

Print["n=",n];
Plot[ (HermiteH[n,x] E^(-x^2/2))^2 ,{x,-6,6},PlotRange->All];
      
,{n,0,3}];
;[s]
2:0,1;59,0;159,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,10,Courier,3,12,0,0,0;
:[font = section; inactive; preserveAspect; startGroup]
Problem 11:
:[font = input; preserveAspect]
Clear["Global`*"];
:[font = input; preserveAspect; startGroup]
sphBessel[n_,z_]= Sqrt[Pi/(2 z)] BesselJ[n+1/2,z]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
((Pi/z)^(1/2)*BesselJ[1/2 + n, z])/2^(1/2)
;[o]
     Pi          1
Sqrt[--] BesselJ[- + n, z]
     z           2
--------------------------
         Sqrt[2]
^*)