fAsianOptions (version 3042.82)

GammaFunctions: Gamma and Related Functions

Description

A collection and description of special mathematical functions. The functions include the error function, the Psi function, the incomplete Gamma function, the Gamma function for complex argument, and the Pochhammer symbol. The Gamma function the logarithm of the Gamma function, their first four derivatives, and the Beta function and the logarithm of the Beta function are part of R's base package. For example, these functions are required to valuate Asian Options based on the theory of exponential Brownian motion.

The functions are:

erf	the Error function,
gamma*	the Gamma function,
lgamma*	the logarithm of the Gamma function,
digamma*	the first derivative of the Log Gamma function,
trigamma*	the second derivative of the Log Gamma function,
tetragamma*	the third derivative of the Log Gamma function,
pentagamma*	the fourth derivative of the Log Gammafunction,
beta*	the Beta function,
lbeta*	the logarithm of the Beta function,
Psi	Psi(x) the Psi or Digamma function,
igamma	P(a,x) the incomplete Gamma function,
cgamma	Gamma function for complex argument,

The functions marked by an asterisk are part of R's base package.

Usage

erf(x)
Psi(x)
igamma(x, a)
cgamma(x, log = FALSE)
Pochhammer(x, n)

Arguments

x

[erf] - a real numeric value or vector. [Psi][*gamma][Pochhammer] - a complex numeric value or vector.

a

a complex numeric value or vector.

n

an integer value n >= 0. A notation used in the theory of special functions for the rising factorial, also known as the rising factorial power, Graham et al. 1994.

log

a logical, if TRUE the logarithm of the complex Gamma function is calculated otherwise if FALSE, the complex Gamma function itself will be calculated.

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.

Artin, E. (1964); The Gamma Function, New York, Holt, Rinehart, and Winston Publishing.

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

Examples

# NOT RUN {
## Calculate Error, Gamma and Related Functions

## gamma -
   # Abramowitz-Stegun: Figure 6.1
   x = seq(-4.01, 4.01, by = 0.011)
   plot(x, gamma(x), ylim = c(-5,5), type = "l", main = "Gamma Function")
   lines(x = c(-4, 4), y = c(0, 0))
     
## Psi -
   # Abramowitz-Stegun: Figure 6.1
   x = seq(-4.01, 4.01, by = 0.011)
   plot(x, Psi(x), ylim = c(-5, 5), type = "l", main = "Psi Function")
   lines(x = c(-4, 4), y = c(0, 0))
   # Note: Is digamma defined for positive values only ?

## igamma -
   # Abramowitz-Stegun: Figure 6.3. 
   gammaStar = function(x, a) { igamma(x,a)/x^a }
   # ... create Figure as an exercise.
  
## igamma -
   # Abramowitz-Stegun: Formula 6.5.12
   # Relation to Confluent Hypergeometric Functions
   a = sqrt(2)
   x = pi
   Re ( (x^a/a) * kummerM(-x, a, 1+a) )
   Re ( (x^a*exp(-x)/a) * kummerM(x, 1, 1+a) )
   pgamma(x,a) * gamma(a)
   igamma(x, a)
 
## cgamma -
   # Abramowitz-Stegun: Tables 6.7
   x = 1
   y = seq(0, 5, by = 0.1); x = rep(x, length = length(y))
   z = complex(real = x, imag = y)
   c = cgamma(z, log = TRUE)
   cbind(y, Re(c), Im(c))
    
## cgamma -
   # Abramowitz-Stegun: Examples 4-8:
   options(digits = 10)
   gamma(6.38); lgamma(56.38)                            # 1/2
   Psi(6.38); Psi(56.38)                                 # 3/4
   cgamma(complex(real = 1, imag = -1), log = TRUE )     # 5
   cgamma(complex(real = 1/2, imag = 1/2), log = TRUE )  # 6
   cgamma(complex(real = 3, imag = 7), log = TRUE )      # 7/8 
# }