
In[1]:= lower[f_]:=Exp[-I phi](-D[f,theta]+I Cot[theta] D[f,phi])               

In[2]:= L2[f_]:=-(Csc[theta]^2*D[f,{phi,2}]+Cot[theta]*D[f,theta] + D[f,{theta, 2}])                                                                        

In[3]:= l5m5=Sin[theta]^5 Exp[I 5 phi]                                          

         (5 I) phi           5
Out[3]= E          Sin[theta]

In[4]:= Simplify[L2[l5m5]/l5m5]                                                 

Out[4]= 30

In[5]:= Simplify[lower[l5m5]]                                                   

             (4 I) phi                      4
Out[5]= -10 E          Cos[theta] Sin[theta]

In[6]:= Simplify[L2[%]/%]                                                       

Out[6]= 30

In[7]:= Simplify[lower[%%]]                                                     

           (3 I) phi                                3
Out[7]= 5 E          (7 + 9 Cos[2 theta]) Sin[theta]

In[8]:= Simplify[L2[%]/%]                                                       

Out[8]= 30

In[9]:= Simplify[lower[%%]]                                                     

             (2 I) phi                                           2
Out[9]= -60 E          (5 Cos[theta] + 3 Cos[3 theta]) Sin[theta]

In[10]:= Simplify[L2[%]/%]                                                      

Out[10]= 30

In[11]:= Simplify[lower[%%]]                                                    

             I phi
Out[11]= 30 E      (15 + 28 Cos[2 theta] + 21 Cos[4 theta]) Sin[theta]

In[12]:= Simplify[L2[%]/%]                                                      

Out[12]= 30

In[13]:= Simplify[lower[%%]]                                                    

Out[13]= -30 (30 Cos[theta] + 35 Cos[3 theta] + 63 Cos[5 theta])

In[14]:= Simplify[L2[%]/%]                                                      

Out[14]= 30

In[15]:= Simplify[%%/LegendreP[5,Cos[theta]]]                                   

Out[15]= -3840

In[16]:= Simplify[lower[%%%]]                                                   

         -450 (2 Sin[theta] + 7 (Sin[3 theta] + 3 Sin[5 theta]))
Out[16]= -------------------------------------------------------
                                  I phi
                                 E

In[17]:= Simplify[lower[%]]                                                     

                                                          2
         -50400 (5 Cos[theta] + 3 Cos[3 theta]) Sin[theta]
Out[17]= --------------------------------------------------
                              (2 I) phi
                             E

In[18]:= Simplify[lower[%]]                                                     

                                                3
         -100800 (7 + 9 Cos[2 theta]) Sin[theta]
Out[18]= ----------------------------------------
                         (3 I) phi
                        E

In[19]:= Simplify[lower[%]]                                                     

                                       4
         -3628800 Cos[theta] Sin[theta]
Out[19]= -------------------------------
                    (4 I) phi
                   E

In[20]:= Simplify[lower[%]]                                                     

                            5
         -3628800 Sin[theta]
Out[20]= --------------------
               (5 I) phi
              E

In[21]:= Simplify[lower[%]]                                                     

Out[21]= 0

In[22]:= Integrate[Integrate[Evaluate[l5m5 /. phi->-phi] l5m5 Sin[theta],{phi,0,2 Pi}],{theta,0,Pi}]                                                                       

              
         1024 Pi
Out[22]= -------
           693

In[23]:= l5m5N=l5m5/Sqrt[%]                                                     


In[24]:= Simplify[lower[l5m5N]]
                                                 
              (4 I) phi      77                       4
         -15 E          Sqrt[--] Cos[theta] Sin[theta]
                             Pi
Out[24]= ----------------------------------------------
                               16


In[25]:= Integrate[Integrate[Evaluate[% /. phi->-phi] % Sin[theta],{phi,0,2 Pi}],{theta,0,Pi}]                                                                             

         
Out[25]= 10

