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By Relton, Frederick Ernest

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It is worth while making a few observations on equation (2). Note that if the middle term is missing, so that the equation is in its normal form, wTe must have a = This enables us to say at sight that if the middle term is missing, the solution must have the factor x*. Con­ versely, a function of the form x*Cn(fixy) satisfies an equation in its normal form and y' is missing. Note further that y cannot be zero; otherwise expressed, the first term in the square bracket cannot be CYLINDER FUNCTIONS 41 absent.

3*5. The normal form and the zeros. The ideas expounded in the previous chapter can now be applied to Bessel’s equation. Its leading coefficient is x2, whence we conclude that no solution can have a repeated zero except possibly at the origin. Otherwise expressed, Cn can have no repeated positive zero and the graph cannot touch the ic-axis except possibly at the origin. Com­ paring Bessel’s equation with y" + y'f(x) + yy(x) = 0, we have, in 2*7, f(x ) = ~, g(x) = l — - X X2 u = e x p {— |J f(x )d x } = a;- *.

A partial verification is afforded by the tables; the first five zeros of «70 are given in the text and the corresponding first five zeros of J x are 3-8317 70155 10-1734 13-3237 16-4706. The verification is left to the reader. Extend the theorem to other orders and verify by the roots of J 3 given in the text. Extend the theorem to the cylinder functions. An application is given later in Chap. VI. 24. If cl9 c2, . . are the successive zeros of Jn(x) when n is greater than J, prove that the interval cr+1 — cr decreases with r increasing.

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