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1: 29.17 Other Solutions
If (29.2.1) admits a Lamé polynomial solution E , then a second linearly independent solution F is given by …
2: 1.13 Differential Equations
The following three statements are equivalent: w 1 ( z ) and w 2 ( z ) comprise a fundamental pair in D ; 𝒲 { w 1 ( z ) , w 2 ( z ) } 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 … …
3: 28.5 Second Solutions fe n , ge n
If a nontrivial solution of Mathieu’s equation with q 0 has period π or 2 π , then any linearly independent solution cannot have either period. … As a consequence of the factor z on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as z ± on . …
4: 14.2 Differential Equations
When μ ν 0 , 1 , 2 , , and μ + ν 1 , 2 , 3 , , 𝖯 ν μ ( x ) and 𝖯 ν μ ( x ) are linearly independent, and when μ 0 they are recessive at x = 1 and x = 1 , respectively. …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. When μ 0 and ν 1 2 , P ν μ ( x ) and 𝑸 ν μ ( x ) are linearly independent, and recessive at x = 1 and x = , respectively. …
5: 3.2 Linear Algebra
To an eigenvalue of multiplicity m , there correspond m linearly independent eigenvectors provided that 𝐀 is nondefective, that is, 𝐀 has a complete set of n linearly independent eigenvectors. …
6: 2.9 Difference Equations
When f 0 2 4 g 0 , there are linearly independent solutions w j ( n ) , j = 1 , 2 , such that …
7: 10.24 Functions of Imaginary Order
and J ~ ν ( x ) , Y ~ ν ( x ) are linearly independent solutions of (10.24.1): …
8: 10.45 Functions of Imaginary Order
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): …
9: 28.29 Definitions and Basic Properties
Let w ( z ) be a solution linearly independent of P ( z ) . …
10: 29.8 Integral Equations
Let w ( z ) be any solution of (29.2.1) of period 4 K , w 2 ( z ) be a linearly independent solution, and 𝒲 { w , w 2 } denote their Wronskian. …