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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 α , β , γ , 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 , with exponent pairs { - 1 2 μ , 1 2 μ } , { - 1 2 μ , 1 2 μ } , and { ν + 1 , - ν } , respectively; compare §2.7(i). …
3: 15.10 Hypergeometric Differential Equation
It has regular singularities at z = 0 , 1 , , 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
Given λ together with the condition (28.29.6), the solutions ± ν of (28.29.9) are the characteristic exponents of (28.29.1). … If ν ( 0 , 1 ) is a solution of (28.29.9), then F ν ( z ) , F - ν ( 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 belonging to the exponent α has the expansion (31.11.1) with … For example, consider the Heun function which is analytic at z = a and has exponent α at . … 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 1 2 , and an irregular singular point at . … (28.2.1) possesses a fundamental pair of solutions w I ( z ; a , q ) , w II ( z ; a , q ) called basic solutions with …
§28.2(iii) Floquet’s Theorem and the Characteristic Exponents
28.2.16 cos ( π ν ) = w I ( π ; a , q ) = w I ( π ; a , - q ) .
Either ν ^ or ν is called a characteristic exponent of (28.2.1). …