What Are the Main Properties of the Digamma Function?

The digamma function, written $\psi(z)$, is the logarithmic derivative of the gamma function. Its identities are useful for simplifying expressions, evaluating special values, and studying the gamma function in complex analysis. Here are several standard properties, along with a worked evaluation using two of them.

Definition and domain

Wherever $\Gamma(z)$ is finite and nonzero, define

$$\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}.$$

The gamma function extends meromorphically to the complex plane and has simple poles at $0,-1,-2,\ldots$. The digamma function has simple poles at those same points, each with residue $-1$. It is finite elsewhere. For positive real $x$, $\psi(x)$ is real-valued.

The logarithmic derivative is useful because differentiating a product becomes addition: if $f$ and $g$ are nonzero, then $(fg)'/(fg)=f'/f+g'/g$. This is why gamma-function identities give corresponding digamma identities after logarithmic differentiation.

Recurrence relation

The gamma recurrence is $\Gamma(z+1)=z\Gamma(z)$. Taking its logarithmic derivative gives

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

for $z$ away from the poles. Equivalently, $\psi(z)=\psi(z+1)-1/z$. Repeated application shifts an argument by an integer; for a positive integer $m$,

$$\psi(z+m)=\psi(z)+\sum_{k=0}^{m-1}\frac{1}{z+k},$$

provided none of the terms is singular. In particular, since $\psi(1)=-\gamma$, where $\gamma$ is the Euler–Mascheroni constant,

$$\psi(n)=H_{n-1}-\gamma\qquad(n=1,2,3,\ldots),$$

with $H_j=1+1/2+\cdots+1/j$ and $H_0=0$.

Reflection formula

The gamma reflection identity is $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$. Logarithmically differentiating it yields

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

for noninteger $z$. This formula gives a difference of two digamma values; it does not say that the values are equal. For example, at $z=1/3$, it gives

$$\psi\!\left(\frac{2}{3}\right)-\psi\!\left(\frac{1}{3}\right)=\frac{\pi}{\sqrt{3}}.$$

Multiplication formula

For a positive integer $m$, differentiating the gamma multiplication formula gives

$$\sum_{k=0}^{m-1}\psi\!\left(z+\frac{k}{m}\right) =m\psi(mz)-m\log m.$$

This holds when the arguments avoid digamma poles. For $m=2$, it reads

$$\psi(z)+\psi\!\left(z+\frac{1}{2}\right)=2\psi(2z)-2\log 2.$$

It is often called the digamma multiplication formula; the case $m=2$ is also known as the duplication formula.

Series representation

For $z$ not equal to a nonpositive integer, a standard convergent series is

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

The paired terms are essential: each difference is of order $1/n^2$ for large $n$, so the series converges. This representation follows, for example, by logarithmically differentiating an infinite-product representation of $\Gamma(z)$. At $z=1$, every summand vanishes, recovering $\psi(1)=-\gamma$.

Worked example: evaluate ψ(1/3) and ψ(2/3)

The reflection formula at $z=1/3$ gives the difference:

$$\psi\!\left(\frac{2}{3}\right)-\psi\!\left(\frac{1}{3}\right)=\frac{\pi}{\sqrt{3}}.$$

Use the multiplication formula with $m=3$ and $z=1/3$ to find the sum. Since $\psi(1)=-\gamma$,

$$\begin{aligned} \psi\!\left(\frac{1}{3}\right)+\psi\!\left(\frac{2}{3}\right)+\psi(1) &=3\psi(1)-3\log 3,\\ \psi\!\left(\frac{1}{3}\right)+\psi\!\left(\frac{2}{3}\right) &=-2\gamma-3\log 3. \end{aligned}$$

Adding and subtracting these equations gives

$$\psi\!\left(\frac{1}{3}\right)=-\gamma-\frac{3}{2}\log 3-\frac{\pi}{2\sqrt{3}},\qquad \psi\!\left(\frac{2}{3}\right)=-\gamma-\frac{3}{2}\log 3+\frac{\pi}{2\sqrt{3}}.$$

The sum comes from multiplication, while the difference comes from reflection. Using both identities determines each value individually.

Common mistakes

• Confusing $\psi$ with $\Gamma$. The digamma function is $\Gamma'/\Gamma$, not the gamma function itself.
• Reversing the recurrence sign. The forward shift is $\psi(z+1)=\psi(z)+1/z$; shifting backward subtracts $1/z$.
• Reversing the reflection difference. With the convention above, $\psi(1-z)-\psi(z)=\pi\cot(\pi z)$.
• Ignoring poles. Formulas involving $\psi(z)$ cannot be evaluated at $z=0,-1,-2,\ldots$; reflection also excludes integer $z$.

These identities provide practical ways to shift arguments, relate reflected values, and evaluate sums of digamma values. For other arguments, the series representation or numerical evaluation can be used once the poles have been ruled out.