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Partial Differentials Notes_Part_12

# Partial Differentials Notes_Part_12 - where h0 = 0 while =...

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where h 0 = 0 , h k = 1 + 1 2 + 1 3 + · · · + 1 k , while γ = lim k → ∞ ( h k log k ) . 5772156649 . . . (12 . 102) is known as Euler’s 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 0 ( x ) 2 π log x, Y m ( x ) ∼ − 2 m ( m 1)! π x m , m > 0 . (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. For example, using the second recurrence formula (12.104) along with (12.98), we can write the Bessel function of order 3 2 in elementary terms: J 3 / 2 ( x ) = dJ 1 / 2 ( x ) dx + 1 2 x J 1 / 2 ( x ) = radicalbigg 2 π bracketleftbigg cos x x 1 / 2 sin x 2 x 3 / 2 bracketrightbigg + radicalbigg

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Partial Differentials Notes_Part_12 - where h0 = 0 while =...

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