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# poles

Poles of expression or function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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)```

### Example 2

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

`poles(tan(x), x = -PI..PI)`

### 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)`

### 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)`

### 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)`

### Example 6

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

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

### 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)`

## 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.