Bessel functions

Author: Klaus Rottbrand©
Date: 15 NOV 2007

Bessel 1. kind of order ν: Jν(r)
Bessel 1. kind of order n: Jn(r)
Modified Bessel 3. kind of order ν: Kν(r)     [Macdonald functions]
Integration und evaluation of g(x;p,r,s,t,u) over x- interval [a,b]
Functions and expressions
Mathematical constants
Physical constants in SI units
Input/Output of f(x;p,r,s,t,u) at x=x0
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Bessel 1. kind of order ν: Jν(r)

Order ν =
Location r =
Number of subintervals: N =
For implementation in automatization: (0, 0) ≤ (ν, r) ≤ (10 , 100 ) and (0, 0) ≤ (ν, r) ≤ (100, 50 )

Bessel 1. kind of integer order n: Jn(r)

Order n =
Location r =
Number of subintervals: N =
For implementation in automatization certainly covered: (0,0) ≤ (ν, r) ≤ (100,100)


Modified Bessel 3. kind of order ν: Kν(r)

Order ν =
Location r =
Upper limit (fixed) b =
Number of subintervals: N =
Certainly covered for implementation in automatization is: (0, 0.1) ≤ (ν, r) ≤ (70, 100) and (0, 2.0) ≤ (ν, r) ≤ (100, 100)

DOWN [For auxiliary calculations]
         
Functions
         
Constants



Integration and evaluation of g(x; p,r,s,t,u) over an x-interval [a,b] :

Input g(x) =
Upper limit b =
Lower limit a =
Parameter p =
Parameter r =
Parameter s =
Parameter t =
Parameter u =
Number of subintervals: N =
Display of N+1 point values (0 = NO): 

DOWN [For auxiliary calculations]
         
Functions
         
Constants




Functions and expressions
abs acos asin atan
acosh(x) = log(x+sqrt(x*x -1))
  [1 <= x]
asinh(x) = log(x+sqrt(x*x +1)) atanh(x) =(log(1-x) - log(1+x))/2
  [0 <= x2 < 1]
cos
cosh(x) = (exp(x) + exp(-x))/2 exp log sin
sinh(x) = (exp(x) - exp(-x))/2 sqr(x) = exp(2 * log(x))     xr=pow(x,r) sqrt tan
tanh(x) = (exp(2*x) -1)/(exp(2*x) +1) 2*p*t*exp(-t*x*t*x)*(1/(1+exp(t*x))-1/(1+exp(t))) p*exp(-r*x*x-2*s*x-t) p*exp(-x*x)
1/p*cos(t*sin(x)-r*x) [r: Integer]
Limits 0 up to p=pi: Bessel Jr(t)
exp(t*log(r))*exp(-r*x+(t-1)*x) [r,t>0]
Limits 0 up to ∞: Γ(t)
1/p*exp(t*cos(x))*cos(r*x) [r: Integer]
Limits 0 up to p=pi: Mod. Bessel Ir(t)
exp(-t*cosh(x))*(cosh(r*x)) [t>0]
Limits 0 up to ∞: Mod. Bessel Kr(t)

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Mathematical constants
π = 3.141592653589793238462643 π/2 = 1.570796326794896619231322 π/3 = 1.047197551196597746154214 π/4 = 0.7853981633974483096156608
3π/2 = 4.7123889803846898576 4π/3 = 4.1887902047863909846 2π = 6.2831853071795864769 4π = 12.566370616359172953
π1/2 = 1.7724538509055160272 π1/3 = 1.4645918875615232630 π1/4 = 1.3313353638003897127 (2π)1/2 = 2.5066282746310005024
1/π1/2 = 0.56418958354775628694 1/π1/3 = 0.68278406325529568146 1/π1/4 = 0.75112554446494248285 1/π = 0.31830988618379067153
π2 = 9.8696044010893586188 1/π3/2 = 0.17958712212516656168 1/(2π)1/2 = 0.39894228040143267793 ln(π) = 1.1447298858494001741
2/π = 0.6366197723675814 2/π1/2 = 1.1283791670955125 πe = 22.459157718361045473 21/2/π = 0.45015815807855303477
e = 2.7182818284590452353 21/2 = 1.4142135623730950488 31/2 = 1.7320508075688772935 51/2 = 2.2360679774997896964
γ = 0.577215664901532860606512 Γ(1/2) = 1.772453850905516 1r in Grad = 57.295779513082320876798155 1o in Rad = 0.017453292519943295769237

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Physical Constants in SI units
Speed of light in vacuum[m/s] c = 2.9979250e+8 Elementary charge[C] e = 1.6021917e-19 Acceleration on the earth [m/s2] g = 9.80665 Gravitational constant[Nm2/kg2] G = 6.6732e-11
Avogadro constant[mol-1] NA = 6.022169e+23 Electron rest mass[kg] me = 9.109558e-31 Boltzmann constant[J/K] k = 1.380622e-23 Planck constant[Js] h = 6.626196e-34



In/Output of f(x; p,r,s,t,u) and evaluation at x = x0 :

Eingabe f(x) =  
Eingabe x0 =
Parameter p =
Parameter r =
Parameter s =
Parameter t =
Parameter u =

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Functions
         
Constants




Remark:
In our case the closed type Newton-Cotes formula with n=10 is used for integration and then summation over N subintervals. The error formula looks correspondingly. It is possible to vary N. Such formulas were taken to evaluate Bessel functions.
Formula: ∫x0x10 g(x) dx = 5h/299376 [16067 (g0 + g10) + 106300 (g1 + g9) - 48525 (g2 + g8) + 272400 (g3 + g7) - 260550 (g4 + g6) + 427368 g5 ] - 1346350/326918592 g(12)(ξ) h13
Literature
Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions, Dover (101972), formula 25.4.20 on page 887 (Integration), Gradshteyn I.S and Ryzhik I.M. Tables of Integrals, Series, and Products ACADEMIC PRESS,62000, formula 8.411.4 on page 902 (Jν), formula 8.432.1 on page 907 (Kν): here with ν > −½.
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