Polygamma function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the polygamma function of order m is a <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic function</a> on the <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 defined as the (m + 1)th <a href="/facts/Derivative_of_the_logarithm/VL7GIwkl">derivative of the logarithm</a> of the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>:

ψ
          
            (
            m
            )
          
        
        (
        z
        )
        :=
        
          
            
              
                d
              
              
                m
              
            
            
              
                d
              
              
                z
                
                  m
                
              
            
          
        
        ψ
        (
        z
        )
        =
        
          
            
              
                d
              
              
                m
                +
                1
              
            
            
              
                d
              
              
                z
                
                  m
                  +
                  1
                
              
            
          
        
        ln
        ⁡
        Γ
        (
        z
        )
        .
      
    
    {\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).}

Thus

ψ
          
            (
            0
            )
          
        
        (
        z
        )
        =
        ψ
        (
        z
        )
        =
        
          
            
              
                Γ
                ′
              
              (
              z
              )
            
            
              Γ
              (
              z
              )
            
          
        
      
    
    {\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}}

holds where ψ(z) is the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a> and Γ(z) is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>. They are <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> on 
 
 
 
 
 C
 
 ∖
 
 
 Z
 
 
 ≤
 0
 
 
 
 
 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}}
 
. At all the nonpositive integers these polygamma functions have a <a href="/facts/Isolated_singularity/GHtn8L4u">pole</a> of order m + 1. The function ψ(1)(z) is sometimes called the <a href="/facts/Trigamma_function/w687IqNw">trigamma function</a>.

The logarithm of the gamma function and the first few polygamma functions in the complex plane<table><tbody><tr><td></td><td></td><td></td></tr><tr><td>ln Γ(z)</td><td>ψ(0)(z)</td><td>ψ(1)(z)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>ψ(2)(z)</td><td>ψ(3)(z)</td><td>ψ(4)(z)</td></tr></tbody></table>

Polygamma function open-in-new

Polygamma function