Digamma function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the digamma function is defined as the <a href="/facts/Logarithmic_derivative/VL7GIwkl">logarithmic derivative</a> of the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>:

ψ
        (
        z
        )
        =
        
          
            
              d
            
            
              
                d
              
              z
            
          
        
        ln
        ⁡
        Γ
        (
        z
        )
        =
        
          
            
              
                Γ
                ′
              
              (
              z
              )
            
            
              Γ
              (
              z
              )
            
          
        
        .
      
    
    {\displaystyle \psi (z)={\frac {\mathrm {d} }{\mathrm {d} z}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.}

It is the first of the <a href="/facts/Polygamma_function/IF6W68aL">polygamma functions</a>. This function is <a href="/facts/Monotonic_function/MjQK0YL2">strictly increasing</a> and <a href="/facts/Concave_function/HxVEBkce">strictly concave</a> on 
 
 
 
 (
 0
 ,
 ∞
 )
 
 
 {\displaystyle (0,\infty )}
 
, and it <a href="/facts/Asymptotic_analysis/kt87qLpu">asymptotically behaves</a> as

ψ
        (
        z
        )
        ∼
        ln
        ⁡
        
          z
        
        −
        
          
            1
            
              2
              z
            
          
        
        ,
      
    
    {\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},}

for complex numbers with large modulus (
 
 
 
 
 |
 
 z
 
 |
 
 →
 ∞
 
 
 {\displaystyle |z|\rightarrow \infty }
 
) in the <a href="/facts/Circular_sector/TqwCaEA5">sector</a> 
 
 
 
 
 |
 
 arg
 ⁡
 z
 
 |
 
 <
 π
 −
 ε
 
 
 {\displaystyle |\arg z|<\pi -\varepsilon }
 
 for any 
 
 
 
 ε
 >
 0
 
 
 {\displaystyle \varepsilon >0}
 
.
The digamma function is often denoted as 
 
 
 
 
 ψ
 
 0
 
 
 (
 x
 )
 ,
 
 ψ
 
 (
 0
 )
 
 
 (
 x
 )
 
 
 {\displaystyle \psi _{0}(x),\psi ^{(0)}(x)}
 
 or Ϝ (the uppercase form of the archaic Greek <a href="/facts/Consonant/81X2SGS6">consonant</a> <a href="/facts/Digamma/6dRK1gDG">digamma</a> meaning <a href="/facts/Gamma/RkmPeRj3">double-gamma</a>).
Gamma.

Digamma function open-in-new

Digamma function