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1: 14.21 Definitions and Basic Properties
§14.21(ii) Numerically Satisfactory Solutions
When ν - 1 2 and μ 0 , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane | ph z | 1 2 π is given by P ν - μ ( z ) and Q ν μ ( z ) . …
2: 11.2 Definitions
§11.2(iii) Numerically Satisfactory Solutions
When z = x , 0 < x < , and ν 0 , numerically satisfactory general solutions of (11.2.7) are given by … When z and ν 0 , numerically satisfactory general solutions of (11.2.7) are given by … When ν 0 , numerically satisfactory general solutions of (11.2.9) are given by …(11.2.17) applies when | ph z | 1 2 π with z bounded away from the origin.
3: 10.25 Definitions
§10.25(iii) Numerically Satisfactory Pairs of Solutions
Table 10.25.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.25.1). …
Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.
Pair Region
4: 2.7 Differential Equations
§2.7(iv) Numerically Satisfactory Solutions
This kind of cancellation cannot take place with w 1 ( z ) and w 2 ( z ) , and for this reason, and following Miller (1950), we call w 1 ( z ) and w 2 ( z ) a numerically satisfactory pair of solutions. … In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere. In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are 1 2 π out of phase.
5: 9.2 Differential Equation
§9.2(iii) Numerically Satisfactory Pairs of Solutions
Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).
Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.
Pair Interval or Region
6: 10.2 Definitions
§10.2(iii) Numerically Satisfactory Pairs of Solutions
Table 10.2.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.2.1) for the stated intervals or regions in the case ν 0 . …
Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
7: 14.2 Differential Equations
§14.2(iii) Numerically Satisfactory Solutions
Hence they comprise a numerically satisfactory pair of solutions2.7(iv)) of (14.2.2) in the interval - 1 < x < 1 . When μ - ν = 0 , - 1 , - 2 , , or μ + ν = - 1 , - 2 , - 3 , , P ν - μ ( x ) and P ν - μ ( - x ) are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. … Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval 1 < x < . With the same conditions, P ν - μ ( - x ) and Q ν μ ( - x ) comprise a numerically satisfactory pair of solutions in the interval - < x < - 1 . …
8: 10.47 Definitions and Basic Properties
§10.47(iii) Numerically Satisfactory Pairs of Solutions
For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols J , Y , H , and ν replaced by j , y , h , and n , respectively. For (10.47.2) numerically satisfactory pairs of solutions are i n ( 1 ) ( z ) and k n ( z ) in the right half of the z -plane, and i n ( 1 ) ( z ) and k n ( - z ) in the left half of the z -plane. …
9: 12.2 Differential Equations
All solutions are entire functions of z and entire functions of a or ν . For real values of z ( = x ) , numerically satisfactory pairs of solutions2.7(iv)) of (12.2.2) are U ( a , x ) and V ( a , x ) when x is positive, or U ( a , - x ) and V ( a , - x ) when x is negative. … In , for j = 0 , 1 , 2 , 3 , U ( ( - 1 ) j - 1 a , ( - i ) j - 1 z ) and U ( ( - 1 ) j a , ( - i ) j z ) comprise a numerically satisfactory pair of solutions in the half-plane 1 4 ( 2 j - 3 ) π ph z 1 4 ( 2 j + 1 ) π . …
10: 9.12 Scorer Functions
§9.12(iv) Numerically Satisfactory Solutions
In , numerically satisfactory sets of solutions are given by …