Power Series Solutions to the Bessel Equation Department of - PowerPoint PPT Presentation

Power Series Solutions to the Bessel Equation Power Series Solutions to the Bessel Equation Department of Mathematics IIT Guwahati RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation The Bessel equation The equation x 2 y ′′ + xy ′ + ( x 2 − α 2 ) y = 0 , (1) where α is a nonnegative constant, is called the Bessel equation. The point x 0 = 0 is a regular singular point. We shall use the method of Frobenius to solve this equation. Thus, we seek solutions of the form ∞ � a n x n + r , y ( x ) = x > 0 , (2) n =0 with a 0 � = 0. RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Differentiation of (2) term by term yields ∞ y ′ = � ( n + r ) a n x n + r − 1 . n =0 Similarly, we obtain ∞ y ′′ = x r − 2 � ( n + r )( n + r − 1) a n x n . n =0 Substituting these into (1), we obtain ∞ ∞ ( n + r )( n + r − 1) a n x n + r + � � ( n + r ) a n x n + r n =0 n =0 ∞ ∞ a n x n + r +2 − α 2 a n x n + r = 0 . � � + n =0 n =0 RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation This implies ∞ ∞ [( n + r ) 2 − α 2 ] a n x n + x r a n x n +2 = 0 . � � x r n =0 n =0 Now, cancel x r , and try to determine a n ’s so that the coefficient of each power of x will vanish. For the constant term, we require ( r 2 − α 2 ) a 0 = 0. Since a 0 � = 0, it follows that r 2 − α 2 = 0 , which is the indicial equation. The only possible values of r are α and − α . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Case I. For r = α , the equations for determining the coefficients are: [(1 + α ) 2 − α 2 ] a 1 = 0 and , [( n + α ) 2 − α 2 ] a n + a n − 2 = 0 , n ≥ 2 . Since α ≥ 0, we have a 1 = 0. The second equation yields a n − 2 a n − 2 a n = − ( n + α ) 2 − α 2 = − n ( n + 2 α ) . (3) Since a 1 = 0, we immediately obtain a 3 = a 5 = a 7 = · · · = 0 . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation For the coefficients with even subscripts, we have − a 0 − a 0 a 2 = 2(2 + 2 α ) = 2 2 (1 + α ) , ( − 1) 2 a 0 − a 2 a 4 = 4(4 + 2 α ) = 2 4 2!(1 + α )(2 + α ) , ( − 1) 3 a 0 − a 4 a 6 = 6(6 + 2 α ) = 2 6 3!(1 + α )(2 + α )(3 + α ) , and, in general ( − 1) n a 0 a 2 n = 2 2 n n !(1 + α )(2 + α ) · · · ( n + α ) . Therefore, the choice r = α yields the solution � � ∞ ( − 1) n x 2 n � y ( x ) = a 0 x α 1 + . 2 2 n n !(1 + α )(2 + α ) · · · ( n + α ) n =1 RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Note: The ratio test shows that the power series formula converges for all x ∈ R . For x < 0, we proceed as above with x r replaced by ( − x ) r . Again, in this case, we find that r satisfies r 2 − α 2 = 0 . Taking r = α , we obtain the same solution, with x α is replaced by ( − x ) α . Therefore, the function y α ( x ) is given by � � ∞ ( − 1) n x 2 n � y α ( x ) = a 0 | x | α 1 + 2 2 n n !(1 + α )(2 + α ) · · · ( n + α ) n =1 (4) is a solution of the Bessel equation valid for all real x � = 0. RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Case II.For r = − α , determine the coefficients from [(1 − α ) 2 − α 2 ] a 1 = 0 and [( n − α ) 2 − α 2 ] a n + a n − 2 = 0 . These equations become (1 − 2 α ) a 1 = 0 and n ( n − 2 α ) a n + a n − 2 = 0 . If 2 α is not an integer, these equations give us a n − 2 a 1 = 0 and a n = − n ( n − 2 α ) , n ≥ 2 . Note that this formula is same as (3), with α replaced by − α . Thus, the solution is given by � � ∞ ( − 1) n x 2 n � y − α ( x ) = a 0 | x | − α 1 + , 2 2 n n !(1 − α )(2 − α ) · · · ( n − α ) n =1 (5) which is valid for all real x � = 0. RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Euler’s gamma function and its properties For s ∈ R with s > 0, we define Γ( s ) by � ∞ t s − 1 e − t dt . Γ( s ) = 0+ The integral converges if s > 0 and diverges if s ≤ 0. Integration by parts yields the functional equation Γ( s + 1) = s Γ( s ) . In general, Γ( s + n ) = ( s + n − 1) · · · ( s + 1) s Γ( s ) , for every n ∈ Z + . Since Γ(1) = 1, we find that Γ( n + 1) = n ! . Thus, the gamma function is an extension of the factorial function from integers to positive real numbers. Therefore, we write Γ( s ) = Γ( s + 1) , s ∈ R . s RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Using this gamma function, we shall simplify the form of the solutions of the Bessel equation. With s = 1 + α , we note that (1 + α )(2 + α ) · · · ( n + α ) = Γ( n + 1 + α ) . Γ(1 + α ) 2 − α Choose a 0 = Γ(1+ α ) in (4), the solution for x > 0 can be written ∞ ( − 1) n � x � x � 2 n � α � J α ( x ) = . 2 n !Γ( n + 1 + α ) 2 n =0 The function J α defined above for x > 0 and α ≥ 0 is called the Bessel function of the first kind of order α . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation When α is a nonnegative integer, say α = p , the Bessel function J p ( x ) is given by ∞ ( − 1) n � x � 2 n + p � J p ( x ) = , ( p = 0 , 1 , 2 , . . . ) . n !( n + p )! 2 n =0 This is a solution of the Bessel equation for x < 0. Figure : The Bessel functions J 0 and J 1 . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation ∈ Z + , define a new function J − α ( x ) (replacing α by − α ) If α / ∞ ( − 1) n � x � x � 2 n � − α � J − α ( x ) = . 2 n !Γ( n + 1 − α ) 2 n =0 With s = 1 − α , we note that Γ( n + 1 − α ) = (1 − α )(2 − α ) · · · ( n − α )Γ(1 − α ) . Thus, the series for J α ( x ) is the same as that for y − α ( x ) in (5) 2 α with a 0 = Γ(1 − α ) , x > 0. If α is not positive integer, J − α is a solution of the Bessel equation for x > 0. ∈ Z + , J α ( x ) and J − α ( x ) are linearly independent on If α / x > 0. The general solution of the Bessel equation for x > 0 is y ( x ) = c 1 J α ( x ) + c 2 J − α ( x ) . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation Useful recurrence relations for J α d dx ( x α J α ( x )) = x α J α − 1 ( x ) . • � � ∞ ( − 1) n � 2 n + α d d � x � dx ( x α J α ( x )) = x α n ! Γ(1 + α + n ) 2 dx n =0 � ∞ � ( − 1) n x 2 n +2 α d � = n ! Γ(1 + α + n )2 2 n + α dx n =0 ∞ ( − 1) n (2 n + 2 α ) x 2 n +2 α − 1 � = . n ! Γ(1 + α + n )2 2 n + α n =0 Since Γ(1 + α + n ) = ( α + n )Γ( α + n ), we have ∞ ( − 1) n 2 x 2 n +2 α − 1 d � dx ( x α J α ( x )) = n ! Γ( α + n )2 2 n + α n =0 ∞ ( − 1) n � 2 n + α − 1 � x � = x α n ! Γ(1 + ( α − 1) + n ) 2 n =0 = x α J α − 1 ( x ) . RA/RKS MA-102 (2016)

Power Series Solutions to the Bessel Equation The other relations involving J α are: d dx ( x − α J α ( x )) = − x − α J α +1 ( x ) . • x J α ( x ) + J ′ α ( x ) = J α − 1 ( x ) . • α x J α ( x ) − J ′ α ( x ) = J α +1 ( x ) . • α • J α − 1 ( x ) + J α +1 ( x ) = 2 α x J α ( x ) . • J α − 1 ( x ) − J α +1 ( x ) = 2 J ′ α ( x ) . Note: Workout these relations. *** End *** RA/RKS MA-102 (2016)