# fundamental pair

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## 8 matching pages

##### 1: 1.13 Differential Equations

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###### Fundamental Pair

►Two solutions ${w}_{1}(z)$ and ${w}_{2}(z)$ are called a*fundamental pair*if any other solution $w(z)$ is expressible as …A fundamental pair can be obtained, for example, by taking any ${z}_{0}\in D$ and requiring that … ►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 … ►If ${w}_{0}(z)$ is any one solution, and ${w}_{1}(z)$, ${w}_{2}(z)$ are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …##### 2: 13.2 Definitions and Basic Properties

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►Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are
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►A fundamental pair of solutions that is numerically satisfactory near the origin is
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►When $b=n+1=1,2,3,\mathrm{\dots}$, a fundamental pair that is numerically satisfactory near the origin is $M(a,n+1,z)$ and
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►When $b=-n=0,-1,-2,\mathrm{\dots}$, a fundamental pair that is numerically satisfactory near the origin is ${z}^{n+1}M(a+n+1,n+2,z)$ and
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##### 3: 28.29 Definitions and Basic Properties

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►If $\nu $
$(\ne 0,1)$ is a solution of (28.29.9), then ${F}_{\nu}(z)$, ${F}_{-\nu}(z)$ comprise a fundamental pair of solutions of Hill’s equation.
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##### 4: 13.14 Definitions and Basic Properties

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►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are
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►A fundamental pair of solutions that is numerically satisfactory in the sector $|\mathrm{ph}z|\le \pi $ near the origin is
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##### 5: 28.2 Definitions and Basic Properties

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►(28.2.1) possesses a fundamental pair of solutions ${w}_{\text{I}}(z;a,q),{w}_{\text{II}}(z;a,q)$ called

*basic solutions*with …##### 6: 15.10 Hypergeometric Differential Equation

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►When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs
${f}_{1}(z)$, ${f}_{2}(z)$ of fundamental solutions.
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►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as

*Kummer’s solutions*. …##### 7: 18.38 Mathematical Applications

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►Classical OP’s play a fundamental role in Gaussian quadrature.
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