# fundamental pair

(0.002 seconds)

## 8 matching pages

##### 1: 1.13 Differential Equations
###### FundamentalPair
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 $D$; $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}$ 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 … 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 $b=n+1=1,2,3,\dots$, a fundamental pair that is numerically satisfactory near the origin is $M\left(a,n+1,z\right)$ and … When $b=-n=0,-1,-2,\dots$, a fundamental pair that is numerically satisfactory near the origin is $z^{n+1}\*M\left(a+n+1,n+2,z\right)$ and …
##### 3: 28.29 Definitions and Basic Properties
If $\nu$ $(\neq 0,1)$ 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 $|\operatorname{ph}{z}|\leq\pi$ near the origin is …
##### 5: 28.2 Definitions and Basic Properties
(28.2.1) possesses a fundamental pair of solutions $w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)$ 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: 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. …
##### 8: 3.11 Approximation Techniques
The pair of vectors $\{\mathbf{f},\mathbf{a}\}$ …is called a discrete Fourier transform pair. … is of fundamental importance in the FFT algorithm. …