Power series expansions for Mathieu functions with small arguments

Kokkorakis, Gerassimos C.; Roumeliotis, John A.

doi:10.1090/s0025-5718-00-01227-8

Cited by 27 publications

(25 citation statements)

References 28 publications

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“…This article fills a gap in the literature on Mathieu functions by extending and improving the results given in [22]. Preliminary results of this research were given in [23].…”

Section: Introductionmentioning

confidence: 67%

“…Power series expansions for Mathieu functions already present in the literature are not useful for the following two reasons. First, published results only contain expansions for the angular functions, but not for the radial functions, such as [1], [26], and [22]. Second, the normalization used is not appropriate.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, various normalizations exist for Mathieu functions and the two most frequently used are the one of Goldstein-Ince [17], [21] and the one of Stratton-Morse-Chu [37,36]. When it comes to small parameter power series expansions, most published results apply the Goldstein-Ince normalization, such as [1], [26], with the exception of [22].…”

Section: Introductionmentioning

confidence: 99%

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions

2009

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Abstract. Small parameter power series expansions for both radial and angular Mathieu functions are derived. The expansions are valid for all integer orders and apply the Stratton-Morse-Chu normalization. Three new contributions are provided: (1) explicit power series expansions for the radial functions, which are not available in the literature; (2) improved convergence rate of the power series expansions of the radial functions, obtained by representing the radial functions as a series of products of Bessel functions; (3) simpler and more direct derivations for the power series expansion for both the angular and radial functions. A numerical validation is also given.

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“…This article fills a gap in the literature on Mathieu functions by extending and improving the results given in [22]. Preliminary results of this research were given in [23].…”

Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions

2009

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“…Application of the method for different stability regions is discussed. Mathieu functions [1][2][3] has simplified theoretical description of the motion of ions in a quadrupole field. As a result of this progress, the algebraic aspects of the Mathieu functions were implemented "simply as another special function".…”

mentioning

confidence: 99%

Ion energy in quadrupole mass spectrometry

Baranov¹

2004

J. Am. Soc. Mass Spectrom.

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Application of an analytical solution of the Mathieu equation in conjunction with algebraic presentation of the Mathieu functions for description of the ion energy in a radiofrequency quadrupole field is discussed in this work. The analytical approach is used to express the ion energy averaged over the initial ion velocity distribution function, field phase and ion residence time. Comparisons with the approximate solutions for potential ion energy are presented with demonstration of their limits. Application of the method for different stability regions is discussed. Mathieu functions [1][2][3] has simplified theoretical description of the motion of ions in a quadrupole field. As a result of this progress, the algebraic aspects of the Mathieu functions were implemented "simply as another special function". The analytical method of the solution of the Mathieu equation in conjunction with algebraic presentation of the Mathieu functions allows introduction of simplifications and generalizations, delivering an alternative method to both the numerical solution of the Mathieu equation and to the matrix method. Previous attempts to use the Mathieu functions were too general due to the absence of simple algorithms or too local due to the necessity to use expansion series around small quantities (see [4] and references therein). The analytical method, in contrast to the matrix method, utilizes a single solution for a complete ion trajectory. The closed formulae obtained provide a general and preferable method for analytical expression of the fundamental properties of the quadrupole field such as ion trajectory stability, transmission/acceptance, resonance (see [5]), and momentum/energy characteristics of the ion motion. The linear quadrupole and the quadrupole trap are considered in this work as examples that demonstrate the advantages of the analytical method for the determination of the fundamental properties of the mass selecting devices as well as their effect on the ion energy. Illustrations of practical implementation of the method are also given for different regions of stability. In this work it is demonstrated that in order to evaluate the average ion energy in the quadrupole RF field, one has to calculate two dimensionless parameters 21 2 ͑a, q͒ and 22 2 ͑a, q͒, which depend only on the field properties. Linear QuadrupoleFor the two-dimensional quadrupole capacitor, the potential of the RF driven quadrupole field can be expressed as a combination of two terms having spatial (U) and periodic (V) trapping potentials:where r 0 is the electrode separation, is the main trapping RF angular frequency and is the initial phase of the main trapping frequency. In the vicinity of the z-axis, the electrostatic field of such a quadrupole has the equipotential lines forming a hyperbolic surface. That leads us to the system of equations for motion of an ion (with mass m and charge e) having arbitrary initial conditions {x i , y i , z i } and {ẋ i , ẏ i , ż i } (i-initial, t i ϭ 0) in Cartesian system of coordinates:Ά mẍ ͑t͒ ϭ ...

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“…However, contrary to the matrix method, the analytical method is limited to the cos trapping waveforms. T he recent development of algebraic methods to compute Mathieu functions [1][2][3] has a potential to simplify significantly the theoretical description of the motion of ions in a quadrupole trap. As a result of this progress the algebraic aspects of Mathieu functions were implemented in computer algebra systems such as Mathematica [4] covering a broad range of a and q parameters.…”

mentioning

confidence: 99%

Analytical approach for description of ion motion in quadrupole mass spectrometer

Baranov¹

2003

J. Am. Soc. Mass Spectrom.

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Power series expansions for Mathieu functions with small arguments

Cited by 27 publications

References 28 publications

New method to obtain small parameter power series expansions of Mathieu radial and angular functions

New method to obtain small parameter power series expansions of Mathieu radial and angular functions

Ion energy in quadrupole mass spectrometry

Analytical approach for description of ion motion in quadrupole mass spectrometer

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