Gamma Functions
- All the Gamma Functions
Stirling's Approximation:
Uses the Definition of Stirling's Approximation. Stirling's Approximation is an approximation for factorials.
Parameters:
n: The number for which the factorial will be approximated via Stirling's Approximation.
Returns:
The approximation of the factorial of the given number.
Equation:
\[ n!\sim \sqrt{2\pi n}\left( \frac{n}{e} \right)^{n} \]
Example:
use ferrate::special::Gamma;
fn main() {
let n = 2_f64;
let stirling = Gamma::stirling(n);
println!("The Stirling Approximation of {}! is: {}", n, stirling);
}
Executes as:
>>> The Stirling Approximation of 2! is: 1.9190043514880554
Log Gamma Lanczos:
The Log Gamma Function is calculated via the Lanczos Approximation. The Lanczos Approximation is an algorithm uses precomputed coefficients to perform calculations for the Gamma and Log-Gamma Function with fixed precision.
Parameters:
n: The number for which the Log Gamma Function will be calculated.
Returns:
The Log Gamma Function of the given number.
Procedure:
Lanczos Approximation for the Log Gamma Function goes as follows:
1. The Constants are defined:
const G: f64 = 5f64;
const P: [f64; 7] = [
1.000000000189712,
76.18009172948503,
-86.50532032927205,
24.01409824118972,
-1.2317395783752254,
0.0012086577526594748,
-0.00000539702438713199,
];
2. A loop calculates for the Final Constant:
let A(z) = P[0] + ∑[k=1; P.len()] P[k] / (z + k)];
3. The Log Gamma Function is calculated:
lnΓ(z) = ln(√(2π)) + ln(A(z)) - (z + G - 0.5) * ln(z + G - 0.5) * (z + 0.5);
Example:
use ferrate::special::Gamma;
fn main() {
let n = 6_f64;
let lanczos_ln = Gamma::lanczosln(n);
println!("The Log Gamma Function of {} is: {}", n, lanczos_ln);
}
Executes as:
>>> The Log Gamma Function of 6 is: 4.787491742782046
Gamma Function Lanczos:
The Gamma function is a generalization of the factorial function to non-integer and complex numbers. It is defined for all non-positive integers. The Gamma Function is calculated via the Lanczos Approximation. The Lanczos Approximation is an algorithm uses precomputed coefficients to perform calculations for the Gamma and Log-Gamma Function with fixed precision.
Parameters:
n: The number for which the Gamma Function will be calculated.
Returns:
The Gamma Function of the given number.
Equation:
\[ \Gamma\left( x \right)=\left( x-1 \right)!\] \[ \Gamma\left( x \right)=\int_{0}^{\infty }t^{x-1}e^{-t}dt\] \[ \Gamma\left( x \right)=e^{ln\Gamma\left( x \right)} \]
Example:
use ferrate::special::Gamma;
fn main() {
let n = 6_f64;
let lanczos = Gamma::lanczos(n);
println!("The Gamma Function of {} is: {}", n, lanczos);
}
Executes as:
>>> The Gamma Function of 6 is: 120
Incomplete Gamma Function (Lower)
Uses the Series Definition of the Lower Incomplete Gamma Function
Parameters:
a: The value for which the Lower Incomplete Gamma Function will be calculated.x: The integral bound at which the Lower Incomplete Gamma Function will be calculated.
Returns:
The Lower Incomplete Gamma Function of the given number.
Equation:
\[ \gamma\left( s,x \right)=\int_{0}^{x }t^{s-1}e^{-t}dt \] \[ \gamma\left( s,x \right)=x^{s}\sum_{k=0}^{\infty }\frac{\left( -1 \right)^{k}x^{k}}{\left( s+k \right)k!} \]
Example:
use ferrate::special::Gamma;
fn main() {
let a = 3_f64;
let x = 1_f64;
let gamma = Gamma::incgamma(a, x);
println!("The Lower Incomplete Gamma Function of {} and {} is: {}", a, x, gamma);
}
Executes as:
>>> The Lower Incomplete Gamma Function of 3 and 1 is: 0.16060279414278839
Incomplete Gamma Function (Upper)
Uses the Definition of the Upper Incomplete Gamma Function.
Parameters:
a: The value for which the Upper Incomplete Gamma Function will be calculated.x: The integral bound at which the Upper Incomplete Gamma Function will be calculated.
Returns:
The Upper Incomplete Gamma Function of the given number.
Equation:
\[ \Gamma\left( s,x \right)=\int_{x}^{\infty }t^{s-1}e^{-t}dt \] \[ \Gamma\left( s,x \right)+\gamma\left( s,x \right)=\Gamma\left( x \right) \] \[ \Gamma\left( s,x \right)=\Gamma\left( x \right)-\gamma\left( s,x \right) \]
Example:
use ferrate::special::Gamma;
fn main() {
let a = 3_f64;
let x = 1_f64;
let gamma = Gamma::incgammac(a, x);
println!("The Upper Incomplete Gamma Function of {} and {} is: {}", a, x, gamma);
}
Executes as:
>>> The Upper Incomplete Gamma Function of 3 and 1 is: 1.8393972058572117
The Regularized Incomplete Gamma Function
Uses the Definition of the Regularized Incomplete Gamma Function.
Parameters:
a: The value for which the Regularized Incomplete Gamma Function will be calculated.x: The integral bound at which the Regularized Incomplete Gamma Function will be calculated.
Returns:
The Regularized Incomplete Gamma Function of the given number.
Equation:
\[ \text{P}\left( s,x \right)=\frac{\gamma\left( s,x \right)}{\Gamma\left( x \right)} \]
Example:
use ferrate::special::Gamma;
fn main() {
let a = 3_f64;
let x = 1_f64;
let gamma = Gamma::reggamma(a, x);
println!("The Regularized Incomplete Gamma Function of {} and {} is: {}", a, x, gamma);
}
Executes as:
>>> The Regularized Incomplete Gamma Function of 3 and 1 is: 0.052653017343711174
CDF for Poisson Random Variables
Uses the Definition of the CDF for Poisson Random Variables.
Parameters:
a: The value for which the CDF for Poisson Random Variables will be calculated.x: The integral bound at which the CDF for Poisson Random Variables will be calculated.
Returns:
The CDF for Poisson Random Variables of the given number.
Equation:
\[ \text{Q}\left( s,x \right)=\frac{\Gamma\left( s,x \right)}{\Gamma\left( x \right)} \] \[ \text{Q}\left( s,x \right)=1-\text{P}\left( s,x \right) \]
Example:
use ferrate::special::Gamma;
fn main() {
let a = 3_f64;
let x = 1_f64;
let gamma = Gamma::reggammac(a, x);
println!("The CDF for Poisson Random Variables of {} and {} is: {}", a, x, gamma);
}
Executes as:
>>> The CDF for Poisson Random Variables of 3 and 1 is: 0.9473469826562888