linearly independent

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11: 31.3 Basic Solutions

… ►When $\gamma \in \mathbb{Z}$, linearly independent solutions can be constructed as in §2.7(i). …

12: 1.2 Elementary Algebra

… ►Nonzero vectors ${𝐯}_1,\mathrm{\dots },{𝐯}_n$ are linearly independent if ${\sum }_{i=1}^n{c}_i{𝐯}_i=𝟎$ implies that all coefficients $c_i$ are zero. A matrix $𝐀$ of order $n$ is non-defective if it has $n$ linearly independent (possibly complex) eigenvectors, otherwise $𝐀$ is called defective. …

13: 3.11 Approximation Techniques

… ►If the functions ${\varphi }_k(x)$ are linearly independent on the set $x_1,x_2,\mathrm{\dots },x_J$, that is, the only solution of the system of equations … ►A set of functions ${\varphi }_0(x),{\varphi }_1(x),\mathrm{\dots },{\varphi }_n(x)$ that is linearly independent on the set $x_1,x_2,\mathrm{\dots },x_J$ (compare (3.11.36)) can always be orthogonalized in the sense given in the preceding paragraph by the Gram–Schmidt procedure; see Gautschi (1997a). …

14: 18.30 Associated OP’s

… ►The corecursive orthogonal polynomials, $p_n^{(0)}(x)$, these being linearly independent solutions of the recurrence for the $p_n(x)$, are defined as follows: … ►Ismail (2009, §2.3) discusses the meaning of linearly independent in this situation. …

15: 21.7 Riemann Surfaces

… ►On a Riemann surface of genus $g$, there are $g$ linearly independent holomorphic differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$. …

16: 2.7 Differential Equations

… ►However, there are unique and linearly independent solutions $w_j(z)$, $j=1,2$, such that …

17: 9.12 Scorer Functions

… ►where $A$ and $B$ are arbitrary constants, $w_1(z)$ and $w_2(z)$ are any two linearly independent solutions of Airy’s equation (9.2.1), and $p(z)$ is any particular solution of (9.12.1). …

18: 15.4 Special Cases

… ►

15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

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19: 19.29 Reduction of General Elliptic Integrals

… ►Basic integrals of type $I(-{𝐞}_j)$, $1\le j\le h$, are not linearly independent, nor are those of type $I({𝐞}_j)$, $1\le j\le 4$. …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. …