Dr. Kamlesh Bisht
                                                                   (Mathematics)
                                                                 Mob. No.-8279829875
                             SPECIAL FUNCTIONS
                UNIT –12: LEGENDRE POLYNOMIAL AND FUNCTIONS
                               Dr. Kamlesh Bisht
                               Academic Consultant
                            Department of Mathematics
                       Uttarakhand Open University, Haldwani
                                                                                       Dr. Kamlesh Bisht
                                                                                        (Mathematics)
                                                                                     Mob. No.-8279829875
                                 CONTENTS
        1.   LEGENDRE’S EQUATION
        1.1. LEGENDRE’S POLYNOMIAL P (x)
                                           n
        1.2. LEGENDRE’S FUNCTION OF THE SECOND KIND i.e. Q (x)
        1.3 GENERAL SOLUTION OF LEGENDRE’S EQUATION                 n
        2. RODRIGUE’S FORMULA
        3. A GENERATING FUNCTION OF LEGENDRE’S     POLYNOMIAL
        4. ORTHOGONALITY OF LEGENDRE POLYNOMIALS
        5. RECURRENCE FORMULAE
        6. NUMERICAL PROBLEMS
        7. LEGENDRE’S POLYNOMIALS APPLICATIONS
        8. NUMERICAL PROBLEMS
                                                   Dr. Kamlesh Bisht
                                                    (Mathematics)
                                                  Mob. No.-8279829875
                 1. LEGENDRE’S EQUATION
                                                                                                               Dr. Kamlesh Bisht
                                                                                                                (Mathematics)
                                                                                                            Mob. No.-8279829875
  Now, 
                                         m             −−12
                                    y = x  (a +a x +a x +...)                                         ...(2)
                                              0     1          2
                                        
                                                mr−
            =                  y          a x
                                        r
                                       r=0
                                   dy       
                                                            m−−r l
            so that                     =−a(m r)
                                   dx      r
                                           r=0
                                     2
                                   dy                                        mr−−2
             and                         =−a(m r)(m−−r l)x
                                   dx2       r
                                            r=0                                                                           
            Substituting these values in (1), we have