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The Mathieu functions MathieuC(a, q, x) and MathieuS(a, q, x) are solutions of the Mathieu differential equation:
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MathieuC and MathieuS are even and odd functions of x, respectively.
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For countably many values of a (as a function of q), MathieuC and MathieuS are 2*Pi-periodic. For , MathieuA(n, q) is the nth such characteristic value for MathieuC, and for , MathieuB(n, q) is the nth characteristic value for MathieuS. The resulting Mathieu functions are:
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where and are normalization constants depending on n and q.
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If the index n is even, then both MathieuCE and MathieuSE are Pi-periodic; they are 2*Pi-periodic otherwise. MathieuCE and MathieuSE are even and odd functions of x, respectively.
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MathieuFloquet(a, q, x) is a Floquet solution of Mathieu's equation. It has the form:
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where is the characteristic exponent and is a Pi periodic function.
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MathieuCPrime, MathieuSPrime, MathieuCEPrime, MathieuSEPrime, and MathieuFloquetPrime are the first derivatives with respect to x of the corresponding Mathieu functions. Note that all higher order derivatives can be written in terms of the 0th and 1st derivatives.
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The odd and even Mathieu functions are related to the Floquet solution via:
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The various Mathieu functions are normalized as follows.
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where P is as given in the definition of the Floquet solution.
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The normalizations of MathieuCE, MathieuSE, and their derivatives coincide with the ones in [1] (see references below), except for , where the function implemented in Maple is equal to the function in reference [1] multiplied by sqrt(2). In this way, we have, in Maple, for instance, MathieuCE(0, 1, 0) = 1 while in [1] the right-hand side is equal to . For details on this normalization, see reference [2].
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MathieuExponent is an inverse to both MathieuA and MathieuB in the following sense.
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For q = 0, the Mathieu functions assume special values:
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