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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: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►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. … ►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 $ℝ$. …

3: 14.2 Differential Equations

… ►When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots }$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ 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 $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ 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 ${𝑸}_{\nu }^{\mu }\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty }$, respectively. …

4: 1.13 Differential Equations

… ►The following three statements are equivalent: $w_1(z)$ and $w_2(z)$ comprise a fundamental pair in $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ 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 … …

5: 3.2 Linear Algebra

… ►To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $𝐀$ is nondefective, that is, $𝐀$ has a complete set of $n$ linearly independent eigenvectors. …

6: 2.9 Difference Equations

… ►When $f_0^2\ne 4g_0$, there are linearly independent solutions $w_j(n)$, $j=1,2$, such that …

7: 10.24 Functions of Imaginary Order

… ►and ${\tilde{J}}_{\nu }\left(x\right)$, ${\tilde{Y}}_{\nu }\left(x\right)$ are linearly independent solutions of (10.24.1): …

8: 10.45 Functions of Imaginary Order

… ►and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ 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 $𝒲\left\{w,w_2\right\}$ denote their Wronskian. …