arccot
Inverse cotangent function
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arccot(x
)
arccot(x)
represents the inverse of the cotangent
function.
The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180^{o}.
arccot
is defined for complex arguments.
Floatingpoint values are returned for floatingpoint arguments. Floatingpoint intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.
If the argument is a rational multiple of I
,
the result is expressed in terms of hyperbolic functions. See Example 2.
The inverse cotangent function is multivalued. The MuPAD^{®} arccot
function
returns the value on the main branch. The branch cut is the interval [ i, i] on
the imaginary axis. Thus, arccot
returns values,
such that y = arccot(x) satisfies $$\frac{\pi}{2}<\Re \left(y\right)\le \frac{\pi}{2}$$ for
any finite complex x.
The cot
function returns explicit values
for arguments that are certain rational multiples of π.
For these values, arccot
returns an appropriate
rational multiple of π on
the main branch. See Example 3.
The values jump when the arguments cross a branch cut. See Example 4.
MuPAD defines arccot
as arccot(x)
= arctan(1/x)
, although arccot
can return
an unevaluated function call and does not rewrite itself in terms
of arctan
. As a consequence of this definition,
the real line crosses the branch cut, and arccot
has
a jump discontinuity at the origin.
The float attributes are kernel functions. Thus, floatingpoint evaluation is fast.
When called with a floatingpoint argument, arccot
is
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
Call arccot
with the following exact and
symbolic input arguments:
arccot(1), arccot(1/sqrt(2)), arccot(5 + I), arccot(1/3), arccot(0), arccot(I/2)
arccot(x), arccot(x + 1), arccot(1/x)
Floatingpoint values are computed for floatingpoint arguments:
arccot(0.1234), arccot(5.6 + 7.8*I), arccot(1.0/10^20)
Floatingpoint intervals are computed for interval arguments:
arccot(4...4), arccot(0...1)
Arguments that are rational multiples of I
are
rewritten in terms of hyperbolic functions:
arccot(5*I), arccot(5/4*I), arccot(3*I)
For other complex arguments unevaluated function calls without simplifications are returned:
arccot(1/2^(1/2) + I), arccot(1  3*I)
Some special values are implemented:
arccot(1), arccot((5  2*5^(1/2))^(1/2)), arccot(3^(1/2)  2)
Such simplifications occur for arguments that are trigonometric images of rational multiples of π:
cot(9/10*PI), arccot(cot(9/10*PI))
The values jump when crossing a branch cut:
arccot(0.5*I + 10^(10)), arccot(0.5*I  10^(10))
On the branch cut, the values of arccot
coincide
with the limit “from the right” for imaginary arguments x
= c*i
where 0 < c < 1
:
limit(arccot(0.5*I  1/n), n = infinity); limit(arccot(0.5*I + 1/n), n = infinity); arccot(0.5*I)
The values coincide with the limit “from the left”
for imaginary arguments x = c*i
where 1
< c < 0
:
limit(arccot(0.5*I  1/n), n = infinity); limit(arccot(0.5*I + 1/n), n = infinity); arccot(0.5*I)
The inverse cotangent function can be rewritten in terms of the logarithm function with complex arguments:
rewrite(arccot(x), ln)
diff
, float
, limit
, taylor
, series
, and other system
functions handle expressions involving the inverse trigonometric functions:
diff(arccot(x^2), x), float(arccos(3)*arccot(5 + I))
limit(arccot(1  sin(x)/x), x = 0)
taylor(arccot(1/x), x = 0)
series(arccot(x), x = 0)

Arithmetical expression or floatingpoint interval.
x