# Algebraic methods to compute Mathieu functions by Frenkel D., Portugal R. PDF

By Frenkel D., Portugal R.

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1Differentiating, c f (x) g(x) A @ 1 A = 0 for all x. we find c1 f (x) + c2 g (x) ≡ 0 also, and hence @ f (x) g (x) c2 The homogeneous system has a nonzero solution if and only if the coefficient matrix is singular, which requires its determinant W [ f (x), g(x) ] = 0. (b) This is the contrapositive of part (a), since if f, g were not linearly independent, then their Wronskian would vanish everywhere. (c) Suppose c1 f (x) + c2 g(x) = c1 x3 + c2 | x |3 ≡ 0. then, at x = 1, c1 + c2 = 0, whereas at x = −1, − c1 + c2 = 0.

B) This is the contrapositive of part (a), since if f, g were not linearly independent, then their Wronskian would vanish everywhere. (c) Suppose c1 f (x) + c2 g(x) = c1 x3 + c2 | x |3 ≡ 0. then, at x = 1, c1 + c2 = 0, whereas at x = −1, − c1 + c2 = 0. Therefore, c1 = c2 = 0, proving linear independence. On the other hand, W [ x3 , | x |3 ] = x3 (3 x2 sign x) − (3 x2 ) | x |3 ≡ 0. 1. Only (a) and (c) are bases. 2. Only (b) is a basis. 3. (a) B @ 0 A , @ 1 A; 2 0 (b) 0 1 0 1 3 1 B4C B4C B C , B C; @ 1A @ 0A 0 (c) 1 0 B B B @ 1 0 1 0 1 −2 −1 1 B C B C 1C C B 0C B0C C, B C , B C.

N, is det Ak /det Ak−1 , where Ak is the k × k upper left submatrix of A with entries aij for i, j = 1, . . , k. A formal proof is done by induction. 20. (a–c) Applying an elementary column operation to a matrix A is the same as applying the elementary row operation to its transpose AT and then taking the transpose of the result. 56 implies that taking the transpose does not affect the de39 terminant, and so any elementary column operation has exactly the same effect as the corresponding elementary row operation.