Documentation

# `poles`

Poles of expression or function

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## Syntax

```poles(`f`, `x`)
poles(`f`, `x` = `a..b`)
poles(`f`, `x`, `options`)
poles(`f`, `x` = `a..b`, `options`)
```

## Description

`poles(f, x)` finds nonremovable singularities of `f`. These singularities are called the poles of `f`. Here, `f` is a function of the variable `x`. See Example 1.

`poles(f, x = a..b)` finds the poles in the interval (`a,b`). See Example 2.

If `poles` cannot find all nonremovable singularities and cannot prove that they do not exist, it returns an unevaluated call. See Example 3.

If `poles` can prove that `f` has no poles (either in the specified interval (`a,b`) or in the complex plane), it returns an empty set. See Example 4.

`a` and `b` must be real numbers or infinities. If you provide complex numbers, `poles` uses an empty interval and returns an empty set.

## Examples

### Example 1

Find the poles of these expressions:

```poles(1/(x - I), x); poles(sin(x)/(x - 1), x)```
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### Example 2

Find the poles of the tangent function in the interval ```(-PI, PI)```:

`poles(tan(x), x = -PI..PI)`
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### Example 3

The tangent function has an infinite number of poles. If you do not specify the interval, `poles` cannot find all of them and, therefore, returns an unevaluated call:

`poles(tan(x), x)`
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### Example 4

If `poles` can prove that the expression or function does not have any poles in the specified interval, it returns an empty set:

`poles(tan(x), x = -1..1)`
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### Example 5

Use `Multiple` to find the poles of this expression and their orders. Restrict the search interval to `(-pi, 10*pi)`:

`poles(tan(x)/(x - 1)^3, x = -PI..PI, Multiple)`
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### Example 6

Use `Residues` to find the poles of this expression and their residues:

`poles(a/x^2/(x - 1), x, Residues)`
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### Example 7

Use `Multiple` and `Residues` to find the poles of this expression and their orders and residues:

`poles(a/x^2/(x - 1), x, Multiple, Residues)`
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## Parameters

 `f` Arithmetical expression representing a function in `x`. `x` `a`, `b` Real numbers (including infinities) that specify the search interval for function poles. If you do not specify the interval ```(a, b)```, then `poles` uses the entire complex plane.

## Options

 `Multiple` When you use this option, `poles` finds the poles of `f` and their orders. It returns a set of lists. Each list contains two entries: the value of a pole and its order. See Example 5. `Residues` When you use this option, `poles` finds the poles of `f` and their residues. It returns a set of lists. Each list contains two entries: the value of a pole and its residue. See Example 6.

## Return Values

Set or set of lists. Without the options, `poles` returns a set containing the values of poles. With `Multiple` or `Residues`, it returns a set of lists. Each list contains the value of a pole and its order or residue, respectively. With both options, `poles` returns a set of lists. Each list contains the value of a pole, its order, and residue.