# Sol-HW10 - Solutions for Assignment 10 Spherical Bessel...

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Solutions for Assignment 10 Math Methods Spherical Bessel Functions Nov. 10, 2010 Fall 2010 Partial Diﬀerential Equations Due Nov. 17, 2010 1 (5). In the class we showed that the lowest order spherical Bessel function is j 0 = sin x x Using the recurrence relations, ﬁnd j 1 ( x ) and j 2 ( x ), and prove that j n ( x ) = ( - 1) n x n ± 1 x ∂x ² n j 0 ( x ) Using the recurrence relation: d dx { x - n J n ( x ) } = - x - n J n +1 ( x ) and substituting n = + 1 / 2, we obtain J +3 / 2 ( x ) x +3 / 2 = - 1 x d dx ³ J +1 / 2 ( x ) x +1 / 2 ´ or j +1 ( x ) x +1 = - 1 x d dx ³ j ( x ) x ´ . Starting with = 0 and applying this formula times, we obtain j ( x ) = x ± - 1 x d dx ² j 0 ( x ) Using this relation, we can get j 1 ( x ) = - j 0 0 ( x ) = sin x x 2 - cos x x and j 2 ( x ) = - x d dx ± j 1 ( x ) x ² = sin x x 3 (3 - x 2 ) - 3 cos x x 2 2 (5). Using the deﬁnition of Neumann function in terms of Bessel functions, show that spherical Neumann function η 0 ( x ) = - cos x x Using the recurrence relations, ﬁnd η 1 ( x ) and η 2 ( x ). η 0 ( x ) = r π 2 x N 1 / 2 ( x ) , where N 1 / 2 ( x ) = cos π 2 J 1 / 2 ( x ) - J - 1 / 2 ( x ) sin π 2 = - J - 1 / 2 ( x )

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and J - 1 / 2 ( x ) = 1 x 1 / 2 d dx h x 1 / 2 J 1 / 2 ( x ) i Substituting these into the original equation, and using deﬁnition of j 0 ( x
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Sol-HW10 - Solutions for Assignment 10 Spherical Bessel...

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