scholarly journals Asymptotic Bessel-function expansions for Legendre and Jacobi functions

2019 ◽  
Vol 60 (1) ◽  
pp. 013501 ◽  
Author(s):  
Loyal Durand
Keyword(s):  
Bessel Function ◽  
Jacobi Functions

scholarly journals Some New Recurrence Relations Concerning Jacobi Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hatem Mejjaoli ◽  
Ahmedou Ould Ahmed Salem
Keyword(s):  
Bessel Function ◽  
Recurrence Relations ◽  
Elementary Functions ◽  
Jacobi Function ◽  
Jacobi Functions ◽  
Asymptotic Relationship

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functionsΦλα,β. Based on an asymptotic relationship between the Jacobi function and the Bessel function, the expression of Bessel function in terms of elementary functions follows as particular cases.

Smooth curve drawing using a Bessel function

2015 ◽  
Author(s):  
Y. Q. Du ◽  
M. J. Pan ◽  
Q. Li ◽  
L. Li
Keyword(s):  
Bessel Function ◽  
Smooth Curve

scholarly journals The Power Losses in Cable Lines Supplying Nonlinear Loads

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1374
Author(s):  
Bartosz Rozegnał ◽  
Paweł Albrechtowicz ◽  
Dominik Mamcarz ◽  
Monika Rerak ◽  
Maciej Skaza
Keyword(s):  
Bessel Function ◽  
Cross Section ◽  
Power Loss ◽  
Sine Wave ◽  
Active Power ◽  
New Method ◽  
Power Losses ◽  
Nonlinear Loads ◽  
Current Harmonics ◽  
Cable Lines

This paper presents the skin effect impact on the active power losses in the sheathless single-core cables/wires supplying nonlinear loads. There are significant conductor losses when the current has a distorted waveform (e.g., the current supplying diode rectifiers). The authors present a new method for active power loss calculation. The obtained results have been compared to the IEC-60287-1-1:2006 + A1:2014 standard method and the method based on the Bessel function. For all methods, the active power loss results were convergent for small-cable cross-section areas. The proposed method gives smaller power loss values for these cable sizes than the IEC and Bessel function methods. For cable cross-section areas greater than 185 mm2, the obtained results were better than those for the other methods. There were also analyses of extra power losses for distorted currents compared to an ideal 50 Hz sine wave for all methods. The new method is based on the current penetration depth factor calculated for every considered current harmonics, which allows us to calculate the precise equivalent resistance for any cable size. This research is part of our work on a cable thermal analysis method that has been developed.

scholarly journals Eisenstein series and an asymptotic for the K-Bessel function

2021 ◽  
Author(s):  
Jimmy Tseng
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Eisenstein Series ◽  
Uniform Estimate ◽  
Positive Real ◽  
Dominant Term ◽  
Complex Order ◽  
Fixed Parameter ◽  
Real Argument ◽  
Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

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