fAsianOptions (version 3042.82)

BesselFunctions: Modified Bessel Functions

Description

A collection and description of special mathematical functions which compute the modified Bessel functions of integer order of the first and second kind as well as their derivatives.

The functions are:

BesselI	modified Bessel function of the 1st Kind,
BesselDI	its derivative,
BesselK	the modified Bessel function of the 3nd Kind,

Usage

BesselI(x, nu, expon.scaled = FALSE) 
BesselK(x, nu, expon.scaled = FALSE)
BesselDI(x, nu) 
BesselDK(x, nu)

Arguments

expon.scaled

a logical; if TRUE, the results are exponentially scaled.

nu

an integer value greater or equal to zero, the integer order of the modified Bessel function.

x

a positive numeric value or a vector of positive numerical values.

Value

The functions return the values of the selected special mathematical function.

References

Abramowitz M., Stegun I.A. (1972); Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover Publishing.

Weisstein E.W. (2004); MathWorld -- A Wolfram Web Resource, http://mathworld.wolfram.com

Examples

# NOT RUN {
## Bessel I0 and K0 - 
   # Abramowitz-Stegun: Table 9.8, p. 416-422
   x = c(0.0, 0.01, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50)  
   data.frame(x, I = exp(-x)*BesselI(x, 0), K = exp(x)*BesselK(x, 0)) 
   # Compare with R's internal function:
   # data.frame(x, ratio = BesselI(x, 0) / besselI(x, 0))
   # data.frame(x, ratio = BesselK(x, 0) / besselK(x, 0))
   
## x = 0:
   c(BesselI(0, 0), BesselI(0, 1), BesselI(0, 2), BesselI(0, 5))
   # Compare with R's internal function:
   # c(besselI(0, 0), besselI(0, 1), besselI(0, 2), besselI(0, 5))
   c(BesselK(0, 0), BesselK(0, 1), BesselK(0, 2), BesselK(0, 5))
   # Compare with R's internal function:
   # c(besselK(0, 0), besselK(0, 1), besselK(0, 2), besselK(0, 5))
   
## Bessel I2 and K2 - 
   # Abramowitz-Stegun: Table 9.8, p. 416-422
   x = c(0.0, 0.01, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50)  
   data.frame(x, I = BesselI(x, 2)/x^2, K = BesselK(x, 2)*x^2) 
   # Compare with R's internal function:
   # data.frame(x, ratio = BesselI(x, 0) / besselI(x, 0))
   # data.frame(x, ratio = BesselK(x, 0) / besselK(x, 0))
   # data.frame(x, ratio = BesselI(x, 1) / besselI(x, 1))
   # data.frame(x, ratio = BesselK(x, 1) / besselK(x, 1))
   # data.frame(x, ratio = BesselI(x, 5) / besselI(x, 5))
   # data.frame(x, ratio = BesselK(x, 5) / besselK(x, 5))
   # data.frame(x, ratio = BesselI(x,50) / besselI(x,50))
   # data.frame(x, ratio = BesselK(x,50) / besselK(x,50))
# }