fundamental pair

SearchAdvancedHelp

(0.001 seconds)

8 matching pages

1: 1.13 Differential Equations

… ►

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 $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ 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

… ►Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory near the origin is … ►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 … ►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 …

3: 28.29 Definitions and Basic Properties

… ►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. …

4: 13.14 Definitions and Basic Properties

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

5: 28.2 Definitions and Basic Properties

… ►(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

… ►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. … ►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: 3.11 Approximation Techniques

… ►The pair of vectors $\{𝐟,𝐚\}$ …is called a discrete Fourier transform pair. … ►is of fundamental importance in the FFT algorithm. …

8: 18.38 Mathematical Applications

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

Integrable Systems

… ►It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. …