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digamma

Represents the digamma function, a special mathematical function derived from the logarithmic derivative of the gamma function.

Overview

Essential in advanced mathematical analysis and number theory, particularly when working with the gamma function and its derivatives.

  • Commonly appears in statistical mechanics and quantum field theory calculations
  • Used in complex analysis for studying properties of meromorphic functions
  • Frequently encountered in mathematical physics research papers and advanced probability theory
  • Important in the study of polygamma functions and related special functions

Examples

Using the digamma function in a functional equation.

ϝ(z+1)=ϝ(z)+1z\digamma(z+1) = \digamma(z) + \frac{1}{z} 
\digamma(z+1) = \digamma(z) + \frac{1}{z}

Expressing the relationship between digamma and harmonic numbers.

ϝ(n+1)+γ=Hn=∑k=1n1k\digamma(n+1) + \gamma = H_n = \sum_{k=1}^n \frac{1}{k} 
\digamma(n+1) + \gamma = H_n = \sum_{k=1}^n \frac{1}{k}

Showing the digamma function's special value at 1.

ϝ(1)=−γ\digamma(1) = -\gamma 
\digamma(1) = -\gamma