# exponent pairs

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## 7 matching pages

##### 1: 15.11 Riemann’s Differential Equation
Here $\{a_{1},a_{2}\}$, $\{b_{1},b_{2}\}$, $\{c_{1},c_{2}\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. …
##### 2: 14.2 Differential Equations
###### §14.2(iii) Numerically Satisfactory Solutions
Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\infty$, with exponent pairs $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, and $\left\{\nu+1,-\nu\right\}$, respectively; compare §2.7(i). …
##### 3: 15.10 Hypergeometric Differential Equation
It has regular singularities at $z=0,1,\infty$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. 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. …
##### 4: 28.29 Definitions and Basic Properties
###### §28.29(ii) Floquet’s Theorem and the Characteristic Exponent
28.29.9 $2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).$
Given $\lambda$ together with the condition (28.29.6), the solutions $\pm\nu$ of (28.29.9) are the characteristic exponents of (28.29.1). … 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. …
##### 5: 36.5 Stokes Sets
They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). …
##### 6: 31.11 Expansions in Series of Hypergeometric Functions
Then the Fuchs–Frobenius solution at $\infty$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\infty$. … Here one of the following four pairs of conditions is satisfied: …
##### 7: 28.2 Definitions and Basic Properties
This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\infty$. … (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 …
###### §28.2(iii) Floquet’s Theorem and the Characteristic Exponents
Either $\widehat{\nu}$ or $\nu$ is called a characteristic exponent of (28.2.1). …