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11—16 of 16 matching pages
11: 28.33 Physical Applications
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►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass per unit area, and radial tension per unit arc length.
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28.33.1
►with , reduces to (28.32.2) with .
…If we denote the positive solutions of (28.33.3) by , then the vibration of the membrane is given by .
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►However, in response to a small perturbation at least one solution may become unbounded.
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12: 1.9 Calculus of a Complex Variable
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►That is, given any positive number , however small, we can find a positive number such that for all in the open disk .
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►For in (), the convergence is absolute and uniform.
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►Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold for all sufficiently small
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13: 19.36 Methods of Computation
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►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated.
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►This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
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►When the values of complete integrals are known, addition theorems with (§19.11(ii)) ease the computation of functions such as when is small and positive.
Similarly, §19.26(ii) eases the computation of functions such as when () is small compared with .
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14: 1.5 Calculus of Two or More Variables
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►that is, for every arbitrarily small positive constant there exists () such that
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►With , , ,
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►Suppose also that converges and
converges uniformly on , that is, given any positive number , however small, we can find a number that is independent of and is such that
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15: 10.73 Physical Applications
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►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain.
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►The functions , , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)):
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10.73.4
►With the spherical harmonic defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
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