/**
 * Mathieu functions: the characteristic values `a_n(q)` / `b_n(q)` and the
 * `2π`-periodic angular functions `ce_n(x, q)` / `se_n(x, q)` of the Mathieu
 * equation
 *
 *     y'' + (a − 2q·cos 2x)·y = 0.
 *
 * Method — the symmetric tridiagonal eigenvalue problem. Expanding the periodic
 * solutions in a Fourier series produces a three-term recurrence among the
 * coefficients, which is a symmetric tridiagonal eigenproblem in each of the
 * four parity classes (DLMF §28.4). The eigenvalues are the characteristic
 * values; the eigenvectors are the Fourier coefficients. We build the truncated
 * matrix and reuse the maintained symmetric eigensolver in
 * `@danielsimonjr/mathts-matrix` (`eig`).
 *
 * Normalization / sign — the standard DLMF / Abramowitz & Stegun convention
 * (identical to `mpmath`/`scipy`): `(1/π)∫₀^{2π} ce_n(x,q)² dx = 1` (so
 * `ce_0(x,0) = 1/√2`, `ce_n(x,0) = cos nx`, `se_n(x,0) = sin nx` for `n ≥ 1`),
 * with the global sign fixed so the dominant Fourier coefficient is positive.
 * Oracle-pinned against `scipy.special` (the installed `mpmath` 1.3.0 lacks the
 * Mathieu family) in `functions/tests/gap-special-mathieu-oracle.test.ts`.
 *
 * These follow the same plain-exported-function pattern as `wave-functions.ts`
 * and `niche.ts` — pure real-valued `number`-in/`number`-out math.
 *
 * @packageDocumentation
 */
/** 64-bit float (default for decimals) */
type f64 = number;
/**
 * Characteristic value `a_n(q)` for the even (cosine-elliptic `ce_n`) Mathieu
 * solutions. As `q → 0`, `a_n → n²`.
 *
 * @param n - Nonnegative integer order.
 * @param q - Mathieu parameter.
 */
export declare function mathieuA(n: number, q: f64): f64;
/**
 * Characteristic value `b_n(q)` for the odd (sine-elliptic `se_n`) Mathieu
 * solutions, `n ≥ 1`. As `q → 0`, `b_n → n²`.
 *
 * @param n - Integer order ≥ 1.
 * @param q - Mathieu parameter.
 */
export declare function mathieuB(n: number, q: f64): f64;
/**
 * Angular Mathieu function `ce_n(x, q)` (cosine-elliptic), summed from its
 * Fourier coefficients. Normalized so `(1/π)∫₀^{2π} ce_n² = 1`; as `q → 0`,
 * `ce_0 → 1/√2` and `ce_n → cos(nx)` for `n ≥ 1`.
 *
 * @param n - Nonnegative integer order.
 * @param q - Mathieu parameter.
 * @param x - Argument in radians.
 */
export declare function mathieuCe(n: number, q: f64, x: f64): f64;
/**
 * Angular Mathieu function `se_n(x, q)` (sine-elliptic), `n ≥ 1`, summed from
 * its Fourier coefficients. Normalized so `(1/π)∫₀^{2π} se_n² = 1`; as `q → 0`,
 * `se_n → sin(nx)`.
 *
 * @param n - Integer order ≥ 1.
 * @param q - Mathieu parameter.
 * @param x - Argument in radians.
 */
export declare function mathieuSe(n: number, q: f64, x: f64): f64;
export {};
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