# linearly independent

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## 1—10 of 19 matching pages

##### 1: 29.17 Other Solutions
If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by …
##### 2: 1.13 Differential Equations
The following three statements are equivalent: $w_{1}(z)$ and $w_{2}(z)$ comprise a fundamental pair in $D$; $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}$ does not vanish in $D$; $w_{1}(z)$ and $w_{2}(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that … …
##### 3: 28.5 Second Solutions $\operatorname{fe}_{n}$, $\operatorname{ge}_{n}$
If a nontrivial solution of Mathieu’s equation with $q\neq 0$ has period $\pi$ or $2\pi$, then any linearly independent solution cannot have either period. … As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to\pm\infty$ on $\mathbb{R}$. …
##### 4: 14.2 Differential Equations
When $\mu-\nu\neq 0,-1,-2,\dots$, and $\mu+\nu\neq-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly independent, and when $\Re\mu\geq 0$ they are recessive at $x=1$ and $x=-1$, respectively. …When $\mu-\nu=0,-1,-2,\dots$, or $\mu+\nu=-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. When $\Re\mu\geq 0$ and $\Re\nu\geq-\frac{1}{2}$, $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\infty$, respectively. …
##### 5: 3.2 Linear Algebra
To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $\mathbf{A}$ is nondefective, that is, $\mathbf{A}$ has a complete set of $n$ linearly independent eigenvectors. …
##### 6: 2.9 Difference Equations
When $f_{0}^{2}\neq 4g_{0}$, there are linearly independent solutions $w_{j}(n)$, $j=1,2$, such that …
##### 7: 10.24 Functions of Imaginary Order
and $\widetilde{J}_{\nu}\left(x\right)$, $\widetilde{Y}_{\nu}\left(x\right)$ are linearly independent solutions of (10.24.1): …
##### 8: 10.45 Functions of Imaginary Order
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): …
##### 9: 28.29 Definitions and Basic Properties
Let $w(z)$ be a solution linearly independent of $P(z)$. …
##### 10: 29.8 Integral Equations
Let $w(z)$ be any solution of (29.2.1) of period $4K$, $w_{2}(z)$ be a linearly independent solution, and $\mathscr{W}\left\{w,w_{2}\right\}$ denote their Wronskian. …