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**Unformatted text preview: **where h = 0 , h k = 1 + 1 2 + 1 3 + + 1 k , while = lim k ( h k log k ) . 5772156649 . . . (12 . 102) is known as Eulers constant . All Bessel functions of the second kind have a singularity at the origin x = 0; indeed, by inspection of (12.101), we find that the leading asymptotics as x 0 are Y ( x ) 2 log x, Y m ( x ) 2 m ( m 1)! x m , m > . (12 . 103) Finally, we show how Bessel functions of different orders are interconnected by two important recurrence relations. Proposition 12.10. The Bessel functions are related by the following formulae : dJ m dx + m x J m ( x ) = J m 1 ( x ) , dJ m dx + m x J m ( x ) = J m +1 ( x ) . (12 . 104) Proof : Let us differentiate the power series x m J m ( x ) = summationdisplay k = 0 ( 1) k x 2 m +2 k 2 2 k + m k ! ( m + k )! . We find d dx [ x m J m ( x )] = summationdisplay k = 0 ( 1) k 2 ( m + k ) x 2 m +2 k 1 2 2 k + m k ! ( m + k )! = x m summationdisplay k = 0 ( 1) k x m 1+2 k 2 2 k + m 1 k ! ( m 1 + k )! = x m J m 1 ( x ) . (12 . 105) Expansion of the left hand side of this formula leads to x m dJ m dx + mx m 1 J m ( x ) = d dx [ x m J m ( x ) ] = x m J m 1 ( x ) , which proves the first recurrence formula (12.104). The second is proved by a similar manipulation involving differentiation of x m J m ( x ). Q.E.D....

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