# Since the number is represented by mathematica as pi

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Unformatted text preview: ). while the mode of oscillation corresponding to the functions f2,1 and g2,1 will vibrate with frequency −λ2,1 α2,1 = = .81736. 2π 2π A general vibration of the vibrating membrane will be a superposition of these modes of oscillation. The fact that the frequencies of oscillation of a circular drum are not integral multiples of a single fundamental frequency (as in the case of the violin string) limits the extent to which a circular drum can be tuned to a speciﬁc tone. Proof that J0 (x) has inﬁnitely many positive zeros: First, we make a change of variables x = ez and note that as z ranges over the real numbers, the corresponding variable x ranges over all the positive real numbers. Since dx = ez dz, d 1d =z dx e dz and hence x 1d d d = ez z = . dx e dz dz Thus Bessel’s equation (5.30) in the case where n = 0 becomes d2 y + e2z y = 0. dz 2 (5.31) Suppose that z0 &gt; 1 and y (z ) is a solution to (5.31) with y (z0 ) = 0. We claim that y (z ) must change sign at some point between z0 and z0 + π ....
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## This document was uploaded on 01/12/2014.

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