Bessel s and legendre s equations
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1 Chapter 12 Bessel s and legendre s equations 12.1 Introduction Many linear differential equations having variable coefficients cannot be solved by usual methods and we need to employ series solution method to find their solutions in terms of infinite convergent series Bessel s Equation The differential equation 1 is known as Bessel s equation of order and it s particular solutions are called Bessel s functions. Series solution of 1 in terms of Bessel s functions and is given by where Proposition If Proof: Case I: is any integer then is a positive integer If is a positive integer, values of from to will give gamma function of integers in the denominator, which being infinite all such terms will vanish., we get Page 1
2 Case II: or Case III:, which is true is a negative integer Let, where is a positive integer From case I Expansions of,, and We have 1. when is a positive integer 2. when is a positive integer 3. Page 2
3 Recurrence Relations of Bessel s Function (1) or Proof: Page 3
4 (2) or Proof: (3) Proof: From recurrence relation (1) Dividing by, we get (4) Proof: From recurrence relation (2) Page 4
5 Dividing by, we get (5) Proof: Adding recurrence relations (3) and (4), we get (6) Proof: Subtracting recurrence relations (3) from (4), we get Example 1 Evaluate,, and Solution: From recurrence relation (6).. 1 in 1, in 1, in 1 Page 5
6 in 1 Example 2 Show that: (i) (ii) (iii) (iv) (v) (vi) Solution: (i), or (ii) From recurrence relation (6).. 1 Page 6
7 Replacing by in 1 (iii) From recurrence relation (5) Replacing in place of in 1.3 Replacing in place of in 1..4 Using 3 and 4 in 2 (iv) From recurrence relation (2) Multiplying both sides by.. 1 Multiplying both sides of 1 by using 1 2 Again multiplying both sides of 2 by Page 7
8 again using 1 Continuing in this manner times, we get (v) 1 From recurrence relation (4) 2 Also from recurrence relation (3)..3 Using 2 and 3 in 1 we get (vi) from (i) Using Differentiating again we get Using Page 8
9 Using Example 3 Show that: (i) (ii) (iii) (iv) (v) Solution: (i) from recurrence relation (1) by putting (ii) (iii) Page 9
10 Integrating by parts Integrating by parts (v) Integrating by parts Integrating by parts Example 4 Express in terms of and Solution: From recurrence relation (6).1 in 1, we get.2 Page 10
11 in 1, we get.3 in 1, we get.4 in 1, we get.5 Using 5 in 4, we get 6 in terms of and Using 6and 5 in 3, we get. 7 in terms of and Using 7 and 6 in 2, we get in terms of and Orthogonality of Bessel s Function If and be roots of the equation, then Proof: Given that and be roots of the equation Page 11
12 1 Consider the Bessel s equations: Solutions of 2 and 3are respectively and Multiplying 2 by and 3by and subtracting Integrating 4 with respect to from 0 to 1 +.4,,,..5 if From 1 Again if, which is form To overcome this difficulty, let be root of the equation, so that, also let Substituting, and taking limit in 5 It is still form, so applying L Hopital s rule on R.H.S. Page 12
13 12.3 Legendre s Equation Another important differential equation used in problems showing spherical symmetry is Legendre s equation given by 1 Here is a real number, though in most practical applications only non-negative integral values are required. Series solution of 1in terms of Legendre s function and is given by, where is a terminating series containing positive powers of. is a non-terminating (infinite) series containing negative powers of Generating function for The function is called the generating function of Legendre s polynomials as Proof: 1 Again 2 Using 2 in 1, we get Page 13
14 3 Coefficient of in expression 3 is given by Corollary (i) (ii) Proof:.. 1 on both sides of 1 Comparing the coefficients of on both sides, we get on both sides of 1 Comparing the coefficients of on both sides, we get Example 5 Prove that Solution: Generating function of is given by..1 Replacing by on both sides of 1 Page 14
15 Again replacing by on both sides of Comparing 2and Recurrence Relations of Legendre s Function (1) Proof: From generating function 1 Differentiating both sides of 1 partially with respect to, we get using 1 Equating coefficient of on both sides (2) Differentiating both sides of 1 partially with respect to, we get using 1 Equating coefficient of on both sides Page 15
16 (3) Differentiating recurrence relation (1) partially with respect to, we get..2 Also from recurrence relation (2) 3 Using 3 in 2, we get (4) Adding recurrence relations (2) and (3), we get (5) Adding recurrence relations (3) and (4), we get (6) Replacing by in recurrence relation (4) 4 Also multiplying recurrence relation (3) by. 5 Subtracting 5 from 4 (7) Replacing by in recurrence relation (3) 6 Page 16
17 Also multiplying recurrence relation (4) by Subtracting 6 from 7, we get.. 7 Example 6 Prove that + C Solution: From recurrence relation (5) Integrating both sides, we get +C Example 7 Show that Solution: From recurrence relation (5), we get Adding all these relations, we get Page 17
18 Rodrigue s Formula Rodrigue s formula is helpful in producing Legendre s polynomials of various orders and is given by Proof: Let, 2 Differentiating 2 ( times using Leibnitz s theorem: 3, so that and 3 which is Legendre s equation with the solution But since constant multiple of contains only positive powers of, solution can only be a. 4, Taking on both sides 5 Page 18
19 Using 5 in 4, we get,,,,,, etc Example 8 Expand the following functions in series of Legendre s polynomials. (i) (ii) Solution:,, (i) Let Substituting values of 1, and in terms of Legendre s polynomials, we get (ii) Let Substituting values of 1,, and in terms of Legendre s polynomials, we get Page 19
20 Example 9 Prove that (i) Solution: (i) (ii), if where and are positive integers Expanding using Leibnitz s theorem 0 on putting or Now when, (ii) By part (i) Continuing the process ( times, we get 0 if i.e Orthogonality of Legendre s polynomial Orthogonality property of Legendre s polynomials is given by the relations, when and, when Page 20
21 where and are positive integers Proof: By Rodrigue s formula and Integrating by parts Continuing the process times 1 using Leibnitz s theorem, when Again putting in 1 Put Page 21
22 Example 10 Prove that (i) (ii) Solution: is the solution of Legendre s equation given by:.1 will satisfy equation 1.2 in 2 we get in 2 we get Exercise 12 A Q1. Prove the following relations for Bessel s Function i. ii. iii. iv. v., Page 22
23 Q2. Prove the following relations for Legendre s Function i. ii. iii. iv., where and are distinct integers Q3. Express the following into Legendre s polynomial: i. Ans. ii. Ans Previous Years Solved Questions Q1. Prove that Solution: Q2. Express the following into Legendre s polynomial: Solution:,, Let Page 23
24 Substituting values of 1,, and in terms of Legendre s polynomials, we get Q3. Prove that Solution: From recurrence relation (6).1 in 1, we get.2 in 1, we get.3 in 1, we get.4 Using 4 in 3, we get 5 in terms of and Using 5 and 4 in 2, we get Q4. Prove that, where and are positive integers Solution: This is orthogonality property of Legendre s polynomials By Rodrigue s formula Page 24
25 and Integrating by parts Continuing the process times 1 using Leibnitz s theorem in 1 Put Page 25
26 Q4. Show that Solution: From recurrence relation (6).. 1 in 1, Q5. Show that Solution: 1 From recurrence relation (4) 2 Also from recurrence relation (3)..3 Using 2 and 3 in 1 we get Page 26
27 Q6. Prove that Solution: From recurrence relation (1) for Legendre s polynomials Replacing by in 1 1 Replacing by in 1 2 Multiplying 2and 3, we get 3 + Integrating both sides w.r.t within the limits 0 to 1 + By orthogonality property Page 27
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