Digamma Function

📐 Definition

For $z \in \mathbb{C} \setminus \{0,-1,-2,\dots\}$, the digamma function is defined as the logarithmic derivative of the Gamma function:

$$\psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} = \frac{d}{dz}\log \Gamma(z)$$

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It provides a natural analytic continuation of harmonic-number–type growth to the complex plane.

Domain & Codomain

Meromorphic on $\mathbb{C}$ with simple poles at the non-positive integers $\{0, -1, -2, \dots\}$

Values in $\mathbb{C}$

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⚙️ Key Properties

Poles and Residues

The poles of $\psi(z)$ occur at the same locations as those of $\Gamma(z)$, namely $z = 0, -1, -2, \dots$

Each pole is simple, with constant residue:

$$\operatorname{Res}\bigl(\psi(z), z = -n\bigr) = -1, \qquad n = 0, 1, 2, \dots$$

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Locally, near a pole $z = -n$, the function behaves as

$$\psi(z) = -\frac{1}{z+n} + \mathcal{O}(1)$$

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Recurrence Relation

The digamma function satisfies a fundamental shift identity inherited from the functional equation of $\Gamma$:

$$\psi(z+1) = \psi(z) + \frac{1}{z}$$

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This relation allows values at large arguments to be reduced recursively to a fundamental domain.

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Reflection Formula

The reflection formula relates values across the line $\Re z = \tfrac12$:

$$\psi(1 - z) - \psi(z) = \pi \cot(\pi z)$$

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This identity is particularly useful for analytic continuation into the left half-plane and exposes the intimate connection between $\psi$ and trigonometric functions.

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Series Representation

For $z \notin \{0,-1,-2,\dots\}$, the digamma function admits the convergent series representation

$$\psi(z) = -\gamma + \sum_{n=0}^{\infty} \left( \frac{1}{n+1} - \frac{1}{n+z} \right)$$

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where $\gamma$ is the Euler–Mascheroni constant.

This form makes explicit the role of $\psi$ as a regularized harmonic sum.

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Asymptotic Behavior in the Complex Plane

As $|z| \to \infty$ with $|\arg z| < \pi$, the digamma function admits the asymptotic expansion

$$\psi(z) = \ln z - \frac{1}{2z} - \frac{1}{12 z^2} + \frac{1}{120 z^4} - \frac{1}{252 z^6} + \mathcal{O}\!\left(z^{-8}\right)$$

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More generally,

$$\psi(z) \sim \ln z - \frac{1}{2z} - \sum_{n=1}^{\infty} \frac{B_{2n}}{2n\, z^{2n}},$$

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where $B_{2n}$ are Bernoulli numbers. The logarithmic term captures the dominant growth, while the inverse powers encode curvature corrections.

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Zeros in the Complex Plane

The digamma function has a unique real zero on the positive real axis:

$$x_0 \approx 1.4616321449683623\ldots$$

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In addition, $\psi(z)$ has infinitely many complex zeros in the left half-plane. These occur in complex-conjugate pairs, cluster near the negative real axis, and asymptotically approach the poles as their imaginary parts increase.

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🎯 Special Values

At selected arguments, the digamma function simplifies to closed-form expressions or well-known constants:

$z$	$\psi(z)$ (exact)	Approximate value
$1$	$-\gamma$	$-0.5772156649$
$2$	$1 - \gamma$	$0.4227843351$
$3$	$\tfrac{3}{2} - \gamma$	$0.9227843351$
$\tfrac{1}{2}$	$-\gamma - 2\ln 2$	$-1.9635100260$
$\tfrac{3}{2}$	$2 - \gamma - 2\ln 2$	$0.0364899740$
$n \in \mathbb{N}$	$H_{n-1} - \gamma$	—

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🔗 Related Functions

Gamma Function Parent function: ψ(z) is the logarithmic derivative of Γ(z)

Trigamma Function First derivative of digamma: ψ'(z)

Polygamma Functions Higher-order derivatives: ψ^(n)(z)

Riemann Zeta Function Appears in analytic continuations and acceleration formulas

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Usage in Oakfield

Applications in Oakfield

Oakfield implements digamma as a core special function and uses it directly in warp/special-function machinery:

• Analytic warp profiles use digamma/trigamma-based gradients (DIGAMMA profile uses trigamma as the derivative of digamma)
• Hyperexponential φ helpers are computed via digamma differences for speed and stability (avoids inner sums and improves numerical behavior)
• Lua exposure: digamma is available through the Lua API for scripts to experiment with warps and special-function diagnostics

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Historical Foundations

1. 📜 Early Foundations (9th-16th Centuries)

   The mathematical foundations underlying the digamma function trace back to work on infinite series and logarithms developed across multiple cultures. Islamic mathematicians including Al-Karaji (c. 953–1029) and Ibn al-Haytham (965–1040) developed sophisticated techniques for summing series and understanding power series expansions.

   The Kerala School of mathematics in India, particularly Madhava of Sangamagrama (c. 1340–1425), derived infinite series expansions for trigonometric and logarithmic functions centuries before their “discovery” in Europe, work later extended by Nilakantha Somayaji (1444–1544).

2. 🔬 European Formalization (18th century)

   Building on these global mathematical traditions, Leonhard Euler (1729) gave the first systematic European treatment of the digamma function in his work on the gamma function and harmonic series. Euler’s formalization drew on the cumulative knowledge of series manipulation and logarithmic derivatives that had developed across continents.

   The notation $\psi$ for the digamma function was later introduced by Carl Friedrich Gauss in the early 19th century.

3. 🌍 Modern Recognition

   Contemporary mathematics increasingly acknowledges that special functions emerged from a global collaborative effort spanning centuries and cultures. The digamma function represents the synthesis of ideas about infinite series, logarithms, and analytic continuation developed independently and interdependently across Islamic, Indian, Chinese, and European mathematical traditions.

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📚 References

• NIST Digital Library of Mathematical Functions (DLMF), §5.15
• Abramowitz & Stegun, Handbook of Mathematical Functions
• Euler, Institutiones Calculi Differentialis (1755)