# uniqueness

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## 1—10 of 49 matching pages

##### 1: 27.15 Chinese Remainder Theorem

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►The Chinese remainder theorem states that a system of congruences $x\equiv {a}_{1}\phantom{\rule{0.949em}{0ex}}(mod{m}_{1}),\mathrm{\dots},x\equiv {a}_{k}\phantom{\rule{0.949em}{0ex}}(mod{m}_{k})$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli.
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►By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod ${m}_{1}$), (mod ${m}_{2}$), (mod ${m}_{3}$), and (mod ${m}_{4}$), respectively.
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##### 2: 28.9 Zeros

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►Furthermore, for $q>0$
${\mathrm{ce}}_{m}(z,q)$ and ${\mathrm{se}}_{m}(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of ${J}_{m}\left(2\sqrt{q}\mathrm{cos}z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im}z\right|\to \mathrm{\infty}$.
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##### 3: Notices

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Index of Selected Software Within the DLMF Chapters
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Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

##### 4: 24.17 Mathematical Applications

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►For each $n$, ${S}_{n}(x)$ is the unique bounded function such that ${S}_{n}(x)\in {\mathcal{S}}_{n}$ and
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►For each $n=1,2,\mathrm{\dots}$ the function ${M}_{n}(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that
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►is the unique cardinal monospline of degree $n$ having the least supremum norm ${\Vert F\Vert}_{\mathrm{\infty}}$ on $\mathbb{R}$ (minimality property).
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##### 5: 2.2 Transcendental Equations

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►Then for $y>f(a)$ the equation $f(x)=y$ has a unique root $x=x(y)$ in $(a,\mathrm{\infty})$, and
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##### 6: 4.12 Generalized Logarithms and Exponentials

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►These functions are not unique.
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##### 7: 2.9 Difference Equations

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►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive.
When $|{\rho}_{2}|=|{\rho}_{1}|$ and $\mathrm{\Re}{\alpha}_{2}=\mathrm{\Re}{\alpha}_{1}$ neither solution is dominant and both are unique.
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►These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e.
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##### 8: 1.8 Fourier Series

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###### Uniqueness of Fourier Series

…##### 9: 2.1 Definitions and Elementary Properties

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►For an example see (2.8.15).
►Asymptotic expansions of the forms (2.1.14), (2.1.16) are unique.
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##### 10: Preface

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►The DLMF has been constructed specifically for effective Web usage and contains features unique to Web presentation.
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