Mathieu function

Although Onsager s first appointment at Yale was a postdoctoral fellowship, he had no doctorate. It disturbed him that everybody called him Dr. Oiisager, and he decided to seek the Ph.D. from Yale. He was told that any of his published works would do for the thesis, but he felt he should write something new, and he quickly submitted a lengthy dissertation on Mathieu functions. Both the Department of Chemistiy and the Department of Physics, found it difficult. The Department of Mathematics, however, was enthusiastic and was prepared to award the degree, whereupon the Department of Chemistiy did not hesitate m accepting the thesis. 

The requirement that i/ be periodic in d with the period 2ir leads to the functions known as Mathieu functions.3 These are usually designated by the symbols ceo, seit cei, se2, ce2, etc. The functions and the corresponding characteristic values of a as functions of q have been evaluated by Goldstein.4 The energy values for the five lowest states are shown in Fig. 3. It is seen... 

E. Mathieu, Liouville s Jour. 13, 137 (1868) Whittaker and Watson, Modern Analysis, pp. 404-428. E. U. Condon, Phys. Rev. 31, 891 (1928), pointed out that the Mathieu functions of even order are the eigenfunctions for the plane pendulum. 

D. W. McLachlan. Theory and Applications of Mathieu Functions. Clarendon Press, Oxford, 1947. 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

Fig. 7. The horizontal projection of the trajectories for the exact solution (the left curve) and the solution of the approximate equation of motion in terms of Mathieu functions (the right curve) iox n = I, k = 2.

Convergence Radii. Applications to Mathieu Functions and to the Two-Electron Problem. 

Application to Mathieu functions. As an application of (5. 3) consider the Mathieu equation... 

M. J. King and J. C. Wiltse, Derivative zeros and other data pertaining to Mathieu functions. Johns Hopkins Radiation Laboratory Technical Report No. AF57, Baltimore, Maryland, 1958. 

National Bureau of Standarts, Tables Relating to Mathieu Function, Columbia Univ. Press, New York, 1951. 

The French mathematician Mathieu investigated this equation in 1868 to describe the vibrations of an elliptical membrane. Mathieu functions are applicable to a wide variety of physical phenomena, e.g problems involving waveguides, diffraction, amplitude distortion, and vibrations in a medium with modulated density. Hill was interested in the motion of planets and was thus engaged in differential equations with periodic integrals [1]. 

To each value of q, a countable infinite set of characteristic values of a is associated for which x t) is an odd or even function that is njr-periodic in time, n being an integer. Series approximations for the characteristic values are obtained by expressing the integral-order Mathieu function as a series of harmonic oscillations, plugging the resultant expression into Eq. (20.8), and equating coefficients of each (orthogonal) frequency component to zero. These laborious calculations yield infinite series in q where each coefficient of q can be expressed as a continued Iraction [4]. 

We observe an exponential decay in the equation. This means that if the Mathieu functions are bounded, the particle will come eventually at rest at the axis x = 0. However, if the Mathieu functions are unbounded, also x(t) is unbounded, as can be verified by inserting appropriate parameters. 

National Bureau of Standards, Computation Laboratory (ed.) Tables Relating to Mathieu Functions Characteristic Values, CoeflBcients, tind Joining Factors. US Department of Commerce, National Bureau of Standards, Washington, DC (1967)... 

As shown in Figure 1 and in many figures in books which describe Mathieu functions [l3,14], the dependence of eigenvalues as a function of q shows an abrupt change in character as they go through a line corresponding to 2q, which is classically the locus of separatrix trajectories. 

J. Meixner, F. W. Schaefke and G. Wolf, Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations, Lecture Notes in Mathematics, Vol. 837 (Springer, Berlin, 1980). 

After replacing in Equation 3.13 by cot 12, the first term of the Mathieu function becomes ... 

The same derivation applies to Equation 3.12 to yield Uy and Qy. Here, one obtains = -Uy and Qjc = -Qy because the x and y components have identical value but opposite polarities. When the amplitude of the ion oscillations in both the x and y directions is anywhere less than Tq, ions are able to propagate through the QME region without colliding with the rods or being pushed away radially. Solving the Mathieu function for stable ion trajectories results in two independent stability criteria (i.e. and ) in the x and y directions. 

Rearranging Equations 3.22 and 3.23, one can obtain a Mathieu function, which is similar to the formula for in QMF (cf. Equation 3.13). In this coordinate system, u represents z or r coordinates, and is still defined as cot 12. Following the same derivation shown in Equation 3.14, Equation 3.22 can be rearranged to yield ... 

Note The Mathieu functions were originally derived in 1868 by Emile Leonard Mathieu, a French mathematician, to describe the vibrations of elliptical drumheads. It turned out that they are also useful to treat quadrupole mass filters and several other physical phenomena. 

Mathieu E (1868) Journal de Mattematiques Pures et Applies (J. Liouville) 13 137. (See also McLachan NW (1947) Theory and Applications of Mathieu Functions. Oxford Clarendon Press and Campbell R (1955) Theo-rie Generate de f Equation de Mathieu. Paris Masson. 

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