The digamma function, denoted as \(\psi(x)\), is a special function defined as the logarithmic derivative of the gamma function. It plays a crucial role in various areas of mathematics, including complex analysis, number theory, and statistics.
The digamma function is defined as:
\[\psi(x) = \frac{d}{dx} \ln(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)}\]
Where \(\Gamma(x)\) is the gamma function.
For positive integers n, the digamma function can be expressed as:
\[\psi(n) = -\gamma + \sum_{k=1}^{n-1} \frac{1}{k}\]
Where \(\gamma\) is the Euler-Mascheroni constant.
\[\psi(x) \approx \ln(x) - \frac{1}{2x} - \frac{1}{12x^2} + \frac{1}{120x^4} - \frac{1}{252x^6} + \cdots\]
\[\psi(x) = \psi(x+1) - \frac{1}{x}\]
Let's calculate \(\psi(5)\):
\[\psi(5) = -\gamma + (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4})\]
\[\psi(5) \approx -0.5772156649 + 2.0833333333 \approx 1.5061176684\]
This graph represents the general shape of the digamma function. It increases monotonically for x > 0 and has poles at non-positive integers.
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