uniqueness

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31: 28.12 Definitions and Basic Properties

… ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization …

32: 29.2 Differential Equations

… ►(These constants are not unique.) …

33: 31.15 Stieltjes Polynomials

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …

34: 1.10 Functions of a Complex Variable

… ►If we can assign a unique value $f(z)$ to $F(z)$ at each point of $D$, and $f(z)$ is analytic on $D$, then $f(z)$ is a branch of $F(z)$. … ►has a unique solution $z=F(w)$ analytic at $w=w_0$, and …

35: 1.2 Elementary Algebra

… ►If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that … ►has a unique solution, $𝐛={𝐀}^{-1}𝐜$. …

36: 2.7 Differential Equations

… ►However, there are unique and linearly independent solutions $w_j(z)$, $j=1,2$, such that … ►The solutions with the properties (2.7.26), (2.7.27) are unique, but not those with the properties (2.7.28), (2.7.29). …

37: 29.6 Fourier Series

… ►When $\nu \ne 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_0,A_2,A_4,\mathrm{\dots }$; furthermore … ►In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. …

38: 3.6 Linear Difference Equations

… ►Then $w_n$ is said to be a recessive (equivalently, minimal or distinguished) solution as $n\to \mathrm{\infty }$, and it is unique except for a constant factor. …

39: 4.2 Definitions

… ►Natural logarithms have as base the unique positive number …

40: 10.41 Asymptotic Expansions for Large Order

… ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. …