On the distribution function of a random power series with Bernoulli variables as coefficients

Abstract In this work, we propose to approximate the Gaussian integral, the error function and the cumulative distribution function by using the power series extender method (PSEM). The approximations proposed in this paper present a high accuracy for the complete domain [–∞,∞]. Furthermore, the approximations are handy and easy computable, avoiding the application of special numerical algorithms. In order to show its high accuracy, three case studies are presented with applications to science and engineering.

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Divergence of random power series.

10.1307/mmj/1028998280 ◽

1959 ◽

Vol 6 (4) ◽

pp. 343-347 ◽

Cited By ~ 2

Author(s):

A. Dvoretzky ◽

P. Erdős

Keyword(s):

Power Series ◽

Random Power Series

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Random power series generated by ergodic transformations

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1986-0854078-9 ◽

1986 ◽

Vol 297 (2) ◽

pp. 461-461 ◽

Cited By ~ 1

Author(s):

Judy Halchin ◽

Karl Petersen

Keyword(s):

Power Series ◽

Ergodic Transformations ◽

Random Power Series

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Inelastic collision integral approximation of the Boltzmann equation

Journal of Plasma Physics ◽

10.1017/s0022377800020006 ◽

1976 ◽

Vol 16 (1) ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Pierre Ségur ◽

Joëlle Lerouvillois-Gaillard

Keyword(s):

Distribution Function ◽

Boltzmann Equation ◽

Power Series ◽

Spherical Harmonic ◽

Inelastic Collision ◽

Collision Integral ◽

Spherical Harmonic Expansion ◽

Square Root ◽

Integral Approximation ◽

The Boltzmann Equation

A study is made of the inelastic collision integral of the Boltzmann equation using scattering probability formalism. The collision operators are expanded in a power series in the square root of the ratio of masses.Furthermore, a spherical harmonic expansion is made of all the operators so obtained. These developments are valid whatever the shape of the distribution function of the particles.

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The distribution of the values of a random power series in the unit disk

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015638 ◽

1983 ◽

Vol 94 (3-4) ◽

pp. 251-263 ◽

Cited By ~ 2

Author(s):

Matthias Jakob ◽

A. C. Offord

Keyword(s):

Power Series ◽

Unit Disk ◽

Random Variables ◽

Equal Probability ◽

Independent Random Variables ◽

Random Power Series ◽

The Family ◽

Unit Radius ◽

Image Position ◽

Almost All

SynopsisThis is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that ifthen almost all these power series take every complex value infinitely often in the unit disk.

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Non-hydrodynamic Contributions to the End Effects in Time of Flight Swarm Experiments

Australian Journal of Physics ◽

10.1071/ph870519 ◽

1987 ◽

Vol 40 (4) ◽

pp. 519 ◽

Cited By ~ 11

Author(s):

Russell K Standish

Keyword(s):

Distribution Function ◽

Power Series ◽

Transport Coefficients ◽

Time Of Flight ◽

Lithium Ions ◽

End Effects ◽

Flight Experiment ◽

Drift Length ◽

Order Of Magnitude ◽

Hydrodynamic Effects

The hydrodynamic part of the distribution function of a swarm is separated from its nonhydrodynamic part using a projection operator, leading to an explicit expression for the timedependent transport coefficients. These are then related to a time of flight experiment. The contribution from non-hydrodynamic effects to the measured drift velocity is shown to be a power series in 11 d, where d is the drift length. A calculation based on an exactly soluble Fokker-Planck model shows that the correction to mobility measurements of lithium ions in helium due to non-hydrodynamic effects is of the same order of magnitude as those observed experimentally.

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