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In mathematics, the Mathieu functions are certain special functions useful for treating a variety of interesting problems in applied mathematics, including

They were introduced by Émile Léonard Mathieu in 1868 in the context of the first problem. Émile Léonard Mathieu ( May 15, 1835, Metz – October 19, 1890, Nancy) was a French Mathematician Year 1868 ( MDCCCLXVIII) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Leap

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Mathieu equation

The canonical form for Mathieu's differential equation is

\frac{d^2u}{dx^2}+[a_u-2q_u\cos (2x) ]u=0

Closely related is Mathieu's modified differential equation

\frac{d^2y}{du^2}-[a-2q\cosh (2u) ]y=0

which follows on substitution u = ix.

The substitution t = cos(x) transforms Mathieu's equation to the algebraic form

(1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0

This has two regular singularities at t = − 1,1 and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function

Mathieu's differential equations arise when the four-dimensional wave equation is written in elliptic cylinder coordinates, followed by a separation of variables. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves Elliptic cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional elliptic coordinate In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an In the algebraic form, it can be seen to be a special case of the spheroidal wave equation. In Mathematics, the spheroidal wave equation is given by (1-t^2\frac{d^2y}{dt^2} -2(b+1 t\ \frac{d y}{dt} + (c - 4qt^2 \ y=0 It is

Floquet solution

According to Floquet's theorem (or Bloch's theorem), for fixed values of a,q, Mathieu's equation admits a complex valued solution of form

F(a,q,x) = \exp(i \mu \,x) \, P(a,q,x)

where μ is a complex number, the Mathieu exponent, and P is a complex valued function which is periodic with period π. Floquet theory is a branch of the theory of Ordinary differential equations relating to the class of solutions to Linear differential equations of the form However, P is in general not sinusoidal. In the example plotted below, a=1, \, q=\frac{1}{5}, \, \mu \approx 1 + 0.0995 i (real part, red; imaginary part; green):

Mathieu sine and cosine

For fixed a,q, the Mathieu cosine C(a,q,x) is a function of x defined as the unique solution of the Mathieu equation which

  1. takes the value C(a,q,0) = 1,
  2. is an even function, hence C^\prime(a,q,0)=0. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

Similarly, the Mathieu sine S(a,q,x) is the unique solution which

  1. takes the value S^\prime(a,q,0)=1,
  2. is an odd function, hence S(a,q,0) = 0. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

These are real-valued functions which are closely related to the Floquet solution:

C(a,q,x) = \frac{F(a,q,x) + F(a,q,-x)}{2 F(a,q,0)}
S(a,q,x) = \frac{F(a,q,x) - F(a,q,-x)}{2 F^\prime(a,q,0)}

The general solution to the Mathieu equation (for fixed a,q) is a linear combination of the Mathieu cosine and Mathieu sine functions.

A noteworthy special case is

C(a,0,x) = \cos(\sqrt{a} x), \; S(a,0,x) = \frac{\sin(\sqrt{a} x)}{\sqrt{a}}

In general, the Mathieu sine and cosine are aperiodic. Nonetheless, for small values of q, we have approximately

C(a,q,x) \approx \cos(\sqrt{a} x), \; \; S(a,q,x) \approx \frac{\sin (\sqrt{a} x)}{\sqrt{a}}

For example:

Red: C(0.3,0.1,x).
Red: C(0. 3,0. 1,x).
Red: C'(0.3,0.1,x).
Red: C'(0. 3,0. 1,x).


Periodic solutions

Given q, for countably many special values of a, called characteristic values, the Mathieu equation admits solutions which are periodic with period 2π. The characteristic values of the Mathieu cosine, sine functions respectively are written a_n(q), \, b_n(q), where n is a natural number. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The periodic special cases of the Mathieu cosine and sine functions are often written CE(n,q,x), \, SE(n,q,x) respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal π). Therefore, for positive q, we have

C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}
S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}

Here are the first few periodic Mathieu cosine functions for q=1:

Note that, for example, CE(1,1,x) (green) resembles a cosine function, but with flatter hills and shallower valleys.

Symbolic computation engines

Various special functions related to the Mathieu functions are implemented in Maple (software) and Mathematica[1]. Maple is a general-purpose commercial Computer algebra system. Mathematica is a computer program used widely in scientific engineering and mathematical fields

See also

References

  1. ^ Mathieu related functions Mathematica documentation. An inverted pendulum is a Pendulum which has its mass above its pivot point In mathematics a Lamé function (or ellipsoidal harmonic function) is a solution of Lamé's equation, a second order ordinary differential equation
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