# linearly independent

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## 1—10 of 19 matching pages

##### 1: 29.17 Other Solutions

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►If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by
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##### 2: 1.13 Differential Equations

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►The following three statements are equivalent: ${w}_{1}(z)$ and ${w}_{2}(z)$ comprise a fundamental pair in $D$; $\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}$ 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 ${\mathrm{fe}}_{n}$, ${\mathrm{ge}}_{n}$

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►If a nontrivial solution of Mathieu’s equation with $q\ne 0$ has period $\pi $ or $2\pi $, then any linearly independent solution cannot have either period.
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►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 $z\to \pm \mathrm{\infty}$ on $\mathbb{R}$.
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##### 4: 14.2 Differential Equations

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►When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots}$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots}$, ${\U0001d5af}_{\nu}^{-\mu}\left(x\right)$ and ${\U0001d5af}_{\nu}^{-\mu}\left(-x\right)$ are linearly independent, and when $\mathrm{\Re}\mu \ge 0$ they are recessive at $x=1$ and $x=-1$, respectively.
…When $\mu -\nu =0,-1,-2,\mathrm{\dots}$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots}$, ${\U0001d5af}_{\nu}^{-\mu}\left(x\right)$ and ${\U0001d5af}_{\nu}^{-\mu}\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.
►When $\mathrm{\Re}\mu \ge 0$ and $\mathrm{\Re}\nu \ge -\frac{1}{2}$, ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty}$, respectively.
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##### 5: 3.2 Linear Algebra

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►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

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►When ${f}_{0}^{2}\ne 4{g}_{0}$, there are linearly independent solutions ${w}_{j}(n)$, $j=1,2$, such that
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##### 7: 10.24 Functions of Imaginary Order

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►and ${\stackrel{~}{J}}_{\nu}\left(x\right)$, ${\stackrel{~}{Y}}_{\nu}\left(x\right)$ are linearly independent solutions of (10.24.1):
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##### 8: 10.45 Functions of Imaginary Order

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►and ${\stackrel{~}{I}}_{\nu}\left(x\right)$, ${\stackrel{~}{K}}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1):
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