# example

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## 1—10 of 179 matching pages

##### 1: 34.9 Graphical Method

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►Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation.
…For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

##### 2: 26.19 Mathematical Applications

##### 3: Guide to Searching the DLMF

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►
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►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function):
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►For example, the expression ${\mathrm{Ai}}^{2}+{\mathrm{Bi}}^{2}$ does not occur verbatim in DLMF, but ${\mathrm{Ai}}^{2}\left(z\right)+{\mathrm{Bi}}^{2}\left(z\right)$ and ${\mathrm{Ai}}^{2}\left(x\right)+{\mathrm{Bi}}^{2}\left(x\right)$ do.
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►For example, for the Bessel function ${K}_{n}\left(z\right)$, you can write

`K_n(z)`

, `BesselK_n(z)`

, `BesselK(n,z)`

, or `BesselK[n,z]`

.
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►For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function.
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##### 4: 26.2 Basic Definitions

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►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$.
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►For an example see Figure 26.9.2.
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►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$.
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►As an example, $\{1,1,1,2,4,4\}$ is a partition of 13.
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►The example
$\{1,1,1,2,4,4\}$ has six parts, three of which equal 1.
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##### 5: 10.76 Approximations

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►Because of the comprehensive nature of more recent software packages (§10.77), the following subsections include only references that give representative examples of the kind of approximations that can be used to generate the functions that appear in the present chapter.
For references to other approximations, see, for example, Luke (1975, §9.13.3).
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##### 6: 27.15 Chinese Remainder Theorem

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►For example, suppose a lengthy calculation involves many 10-digit integers.
…Choose four relatively prime moduli ${m}_{1},{m}_{2},{m}_{3}$, and ${m}_{4}$ of five digits each, for example
${2}^{16}-3$, ${2}^{16}-1$, ${2}^{16}+1$, and ${2}^{16}+3$.
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##### 7: 9.16 Physical Applications

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►Examples dealing with the propagation of light and with radiation of electromagnetic waves are given in Landau and Lifshitz (1962).
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►These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability.
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►An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of $\mathrm{Ai}\left(x\right)$.
…This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
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##### 8: 12.16 Mathematical Applications

##### 9: 25.17 Physical Applications

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►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (

*Casimir–Polder effect*). …