On confluent hypergeometric functions and generalized Bessel functions

2017 ◽  
Vol 43 (4) ◽  
pp. 533-545 ◽  
Author(s):  
N. Bohra ◽  
V. Ravichandran
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Generalized Bessel Functions

scholarly journals Integrals of Lommel's Type for Confluent Hypergeometric Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 28-31
Author(s):  
D. Martin
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Confluent Hypergeometric Functions

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 270-283 ◽  
Keyword(s):  
Exponential Type ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Modified Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Parabolic Cylinder Functions

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE WHITTAKER FUNCTIONS Mκ,iμ(z) AND Wκ,iμ(z)

2003 ◽  
Vol 01 (02) ◽  
pp. 199-212 ◽  
Author(s):  
T. M. DUNSTER
Keyword(s):  
Hypergeometric Functions ◽  
Turning Point ◽  
Bessel Functions ◽  
Asymptotic Approximations ◽  
Whittaker Functions ◽  
Modified Bessel Functions ◽  
Airy Functions ◽  
Double Pole ◽  
Confluent Hypergeometric Functions ◽  
Uniform Reduction

Uniform asymptotic approximations are obtained for the Whittaker's confluent hypergeometric functions Mκ, iμ(z) and Wκ, iμ(z), where κ, μ and z are real. Three cases are considered, and when taken together, result in approximations which are valid for κ → ∞ uniformly for 0 ≤ μ < ∞, 0 < z < ∞, and also for μ → ∞ uniformly for 0 ≤ κ < ∞, 0 < z < ∞. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.

scholarly journals Some Bessel Function Integrals

1956 ◽  
Vol 2 (4) ◽  
pp. 183-184 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Bessel Function ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Basic Formula ◽  
Confluent Hypergeometric Functions ◽  
Image Position

The basic formula to be proved iswhere p≧q + 1, z ≠0; | amp z | < π, R(n)>0, r = 1, 2,…,p. For other values of pand qthe result holds if the integral converges. From this formula some results, involving Bessel functions and Confluent Hypergeometric functions, will be deduced.

Zur elastischen Streuung langsamer Elektronen im Feld eines Atoms angewandt auf die p-Streuung eines Elektrons im Feld des atomaren Wasserstoffs

1959 ◽  
Vol 14 (2) ◽  
pp. 172-193
Author(s):  
F. B. Malik
Keyword(s):  
Wave Scattering ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Quadratic Equation ◽  
Wave Functions ◽  
P Wave ◽  
Hartree Fock ◽  
Confluent Hypergeometric Functions ◽  
Coulomb Type ◽  
Slow Electrons

A method has been developed to calculate the scattering of slow electrons. This method along with the wellknown methods of HULTHÈN and KOHN are applied to calculate p-wave scattering in the normal hydrogen field. The new method avoids the ambiguity of HULTHEN'S quadratic equation and gives almost the same result. The interrelation of the three methods is studied and a proposal is being made how one can simultaneously satisfy the conditions L=∫ψ(H-E) ψ dτ=0, ∂, a k=—〈Rl⎜U⎟jl+〉 appearing in the three methods. The whole theoretical discussion can without difficulties be extended to the modified COULOMB potential, only the regular and singular spherical BESSEL functions are to be substituted by the regular and singular confluent hypergeometric functions of the COULOMB type respectively. The phase shifts and the wave functions calculated without exchange agree well with the numerically solved results of CHANDRASEKHAR and BREEN. The results including exchange terms agree reasonably well with the results of the numerical integration of the HARTREE-FOCK equation, which has been carried out here by E. TREFFTZ.

scholarly journals INEQUALITIES FOR SOME SPECIAL FUNCTIONS-II

1995 ◽  
Vol 26 (3) ◽  
pp. 235-242
Author(s):  
S. K. BISSU ◽  
C. M. JOSHI
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Upper Bounds ◽  
Modified Bessel Functions ◽  
Lower And Upper Bounds ◽  
Confluent Hypergeometric Functions

Some inequalities for Bessel functions, modified Bessel functions of the first kind and of their ratios involving both lower and upper bounds are given. The inequalities improve the results of earlier authours. Also incorporated in the discussion are some inequalities for the ratios of confluent hypergeometric functions of one variable.

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