Calculation of the complex zeros of the modified bessel function of the second kind and its derivatives

“…Since K −ν = K ν and there are no zeros when 0 ≦ ν < 3 2 (see [3]), we throughout assume ν ≧ 3 2 . When ν = 2n + 3 2 , n = 0, 1, ..., K 2n+ 3 2 is of the form π/(2z)z −(2n+1) ϕ n (z), ϕ n (z) being a polynomial of order 2n + 1 (see (2) below). ϕ n has a unique negative root and we regard it as a zero of K 2n+ 3 2 .…”

mentioning

confidence: 99%

On the zeros of the Macdonald functions

Hamana¹,

Matsumoto

Shirai³

2019

Opuscula Math.

View full text Add to dashboard Cite

show abstract

“…This value of h is also confirmed to 9 significant digits by Kerimov and Ckorokhodov (1984). Writing cothðk n lÞ ¼ 1 þ g n ; g n ¼ expðÀk n lÞ sinhðk n lÞ ;…”

Section: Reducing the Problem To A Fredholm Integral Equation Of The mentioning

confidence: 77%

Torsion of a cylinder with a shallow external crack

Malits

2009

International Journal of Solids and Structures

View full text Add to dashboard Cite

a b s t r a c tThe torsional problem of a finite elastic cylinder with a circumferential edge crack is studied in this paper. An efficient solution to the problem is achieved by using a new form of regularization applied to dual Dini series equations. Unlike the Srivastav approach, this regularization transforms dual equations into a Fredholm integral equation of the second kind given on the crack surface. Hence, exact asymptotic expansions of the Fredholm equation solution, the stress intensity factor and the torque are derived for the case of a shallow crack. The asymptotic expansions are certain power-logarithmic series of the normalized crack depth. Coefficients of these series are found from recurrent relations. Calculations for a shallow crack manifest that the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter.

show abstract

“…The values of x0 and y0 with an accuracy of nine significant digits after the decimal point are equal [9]:…”

Section: Inversion Of Laplace's Imagesmentioning

confidence: 99%

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

Filippov

2023

Preprint

View full text Add to dashboard Cite

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite thin elastic plate with a circular orifice of radius lying on a rough base. In the field of motion, an exact analytical solution of a nonlinear boundary value problem for tangential stresses and transversal velocities is obtained in quadratures by the method of Laplace transformations. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance from the center of the hole and an elementary formula is given for this distance (the case of tangential forces instantly applied to the orifice boundary, and then constant in time, is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the trailing wave fronts are obtained, after which a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual deformations is established in the region .

show abstract

Calculation of the complex zeros of the modified bessel function of the second kind and its derivatives

Cited by 8 publications

References 9 publications

On the zeros of the Macdonald functions

On the zeros of the Macdonald functions

Torsion of a cylinder with a shallow external crack

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

Contact Info

Product

Resources

About