On the Computation of Mathieu Functions

Blanch, Gertrude

doi:10.1002/sapm19462511

Cited by 30 publications

(14 citation statements)

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“…There are two standard procedures for finding such solutions. The first was developed independently by Bouwkamp [14] and Blanch [15], and makes use of a fundamental equivalence between three-term recurrences and continued fractions [16]. This provides a method for determining the eigenvalues numerically to high precision, although it relies on the availability of a sufficiently accurate starting estimate for the eigenvalue.…”

Section: Numerical Computation 31 the Spheroidal Eigenvalues L mentioning

confidence: 99%

Theory and computation of spheroidal wavefunctions

Falloon

Abbott

Wang

2003

J. Phys. A: Math. Gen.

123

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Section: Numerical Computation 31 the Spheroidal Eigenvalues L mentioning

confidence: 99%

Theory and computation of spheroidal wavefunctions

Falloon

Abbott

Wang

2003

J. Phys. A: Math. Gen.

123

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“…We use (8) to obtain angular function values for c imaginary from the corresponding values calculated for c real.…”

Section: The Modified Mathieu Functions Of Integer Ordermentioning

confidence: 99%

“…For example, we can use the relations in (8) to replace the function ce n (0, i|c|) appearing in the expression for Mc (1) n with n even by its equivalent (−1) n/2 ce n (π/2, |c|), which is accurately computed from (2). However, the radial functions of the first kind cannot be accurately calculated using these expressions when c is imaginary and large unless n is small because of significant subtraction errors occurring in the series.…”

Section: The Modified Mathieu Functions Of Integer Ordermentioning

confidence: 99%

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Accurate calculation of the modified Mathieu functions of integer order

Buren

Boisvert

2007

Quart. Appl. Math.

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Abstract. Several different expressions involving infinite series are available for calculating the radial (i.e., modified) Mathieu functions of integer order. Mathieu functions depend on three parameters: order, radial coordinate, and a size parameter that is chosen here to be either real or imaginary. Expressions traditionally used to calculate the radial functions result in inaccurate function values over some parameter ranges due to unavoidable subtraction errors that occur in the series evaluation. For many of the expressions the errors for small orders increase without bound as the size parameter increases. In the present paper, the subtraction error obtained using traditional expressions is explored with regard to parameter values. Included is a discussion of the Bessel function product series, which has an integer offset for the order of the Bessel functions that is traditionally chosen to be zero (or one). It is shown here that the use of larger offset values that tend to increase with increasing radial function order usually eliminates the subtraction errors. This paper identifies the expressions and evaluation procedures that provide accurate radial Mathieu function values. A brief discussion of the calculation of the angular functions of the first kind that appear in many of these expressions is included. The paper also gives a description of a Fortran computer program that provides accurate values of radial Mathieu functions together with the associated angular functions over extremely wide parameter ranges. This effort was guided by recent advancements in the calculation of prolate spheroidal functions. The Mathieu functions are a special case of spheroidal functions, resulting in a similarity of behavior in their evaluation.

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“…Methods have been developed for obtaining the characteristic values of linear second-order differential equations such as Mathieu's equation and that of spheroidal wave functions! In a recent paper Miss Blanch (5) has given a method for correcting the characteristic values of Mathieu's equation, and the coefficients, occurring in the Fourier series developments of the characteristic functions. According to Miss Blanch "there does not seem to appear in the literature any method for improving the accuracy of the characteristic values, except by cumbersome iteration.…”

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confidence: 99%

On Spheroidal Wave Functions of Order Zero

Bouwkamp

1947

Journal of Mathematics and Physics

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On the Computation of Mathieu Functions

Cited by 30 publications

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Theory and computation of spheroidal wavefunctions

Theory and computation of spheroidal wavefunctions

Accurate calculation of the modified Mathieu functions of integer order

On Spheroidal Wave Functions of Order Zero

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