Poles of expression or function
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poles(f
,x
) poles(f
,x
=a..b
) poles(f
,x
,options
) poles(f
,x
=a..b
,options
)
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.
Find the poles of these expressions:
poles(1/(x  I), x); poles(sin(x)/(x  1), x)
Find the poles of the tangent function in the interval (PI,
PI)
:
poles(tan(x), x = PI..PI)
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)
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)
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)
Use Residues
to find the poles of this expression
and their residues:
poles(a/x^2/(x  1), x, Residues)
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)

Arithmetical
expression representing a function in 
 

Real numbers (including infinities) that specify the search
interval for function poles. If you do not specify the interval 

When you use this option, See Example 5. 

When you use this option, See Example 6. 
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.