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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
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