Polygamma

The polygamma functions, denoted as (\( \psi^{n}\left( z \right) \)), are mathematical functions that are extensions of the digamma function (\( \psi\left( z \right) \)), also known as the psi or di-gamma function. The digamma function is the logarithmic derivative of the gamma function (\( \Gamma\left( z \right) \)), which is a widely used function in mathematics and has applications in various fields.

Digamma:

Calculates the Digamma function with its series representation. The Digamma function is the logarithmic derivative of the gamma function (\( \Gamma\left( z \right) \)).

Parameters:

z: The number for which the Digamma function will be calculated.

Returns:

The Digamma function of the given number.

Equation:

\[ \psi\left( z \right)=\frac{d }{dz}\Gamma\left( z \right)=\frac{\Gamma'\left( z \right)}{\Gamma\left( z \right)} \]

Example:

use ferrate::special::Polygamma;         

fn main() {                               
    let z = 2_f64;                        
                                          
    let digamma = Polygamma::digamma(z);  
                                          
    println!("The Digamma Function of {} is: {}", z, digamma);
}

Executes as:

>>> The Digamma Function of 2 is: 0.42278438084235914

Polygamma:

Calculates the Polygamma function with its series representation. The Polygamma function is the (\( m \))-th derivative of the Digamma function (\( \psi\left( z \right) \)).

Parameters:

degree: The degree of the Polygamma function. This is the (\( m \)) in (\( \psi^{m}\left( z \right) \)).
z: The number for which the Polygamma function will be calculated.

Returns:

The Polygamma function of the given number.

Equation:

\[ \psi^{\left( m \right)}\left( z \right):=\frac{d^{m}}{dz^{m}}\psi\left( z \right)=\left( -1 \right)^{\left( m+1 \right)}m!\sum_{k=0}^{\infty }\frac{1}{\left( z+k \right)^{m+1}}\qquad m\gt 1 \]

\[ \psi^{-1}\left( z \right)=\int\psi\left( z \right)=\text{ln}\Gamma\left( z \right) \]

Example:

use ferrate::special::Polygamma;                
               
fn main() {                                  
    let degree = 0;                                 
    let z = 5_f64;                                  
                                                
    let polygamma = Polygamma::polygamma(degree, z);

    println!("The Polygamma Function of {} at degree {} is: {}", z, degree, polygamma);                                                
}

Executes as:

>>> The Polygamma Function of 5 at degree 0 is: 1.5061177964312848

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