Singularities

Poles of Order 1 (Simple Poles)

Function 1/(1–z)

The function $\dfrac{1}{1-z} \; (z\in\mathbb{C})$ has one simple pole of order 1 at $z=1$ .
Its finite value at its simple pole $z=1$ is:

$\dfrac{1}{1-1} = -\dfrac{1}{2}$

Function 1/(1–z²)

The function $\dfrac{1}{1-z^2} \; (z\in\mathbb{C})$ has one simple pole of order 1 at $z=1$ .
Its finite value at its simple pole $z=1$ is:

$\dfrac{1}{1-1^2} = 0$

Function 1/(1–z^r)

The function $\dfrac{1}{1-z^r} \; (z\in\mathbb{C} \text{ and } r\in\mathbb{N})$ has one simple pole of order 1 at $z=1$ .
Its finite value at its simple pole $z=1$ is:

$\dfrac{1}{1-1^r} = \dfrac{r-2}{2r}$

Gamma function Γ(z)

The Gamma function $\Gamma(z) \; (z\in\mathbb{C})$ has a simple pole at every non-positive integer. Its finite values at its simple poles are:

$\Gamma(1-n) = (-1)^{n-1} \, \dfrac{\gamma}{\Gamma(n)} = (-1)^{n-1} \, \dfrac{\gamma}{(n-1)!} \qquad (n\in\mathbb{N})$

In particular, $\Gamma(0) = \gamma$ .

Inverse sine function π/sin(πz)

The normalised inverse sine function $\dfrac{\pi}{\sin(\pi z)} \; (z\in\mathbb{C})$ has a simple pole at every integer. Its finite values at its simple poles are: