Inverse cotangent function
MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.
MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.
arccot(x) represents the inverse of the cotangent
The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180o.
arccot is defined for complex arguments.
Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.
If the argument is a rational multiple of
the result is expressed in terms of hyperbolic functions. See Example 2.
The inverse cotangent function is multivalued. The MuPAD®
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 for
any finite complex x.
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.
= arctan(1/x), although
arccot can return
an unevaluated function call and does not rewrite itself in terms
arctan. As a consequence of this definition,
the real line crosses the branch cut, and
a jump discontinuity at the origin.
The float attributes are kernel functions. Thus, floating-point evaluation is fast.
When called with a floating-point argument,
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
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)
Floating-point values are computed for floating-point arguments:
arccot(0.1234), arccot(5.6 + 7.8*I), arccot(1.0/10^20)
Floating-point intervals are computed for interval arguments:
Arguments that are rational multiples of
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:
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
with the limit “from the right” for imaginary arguments
= 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
< 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:
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 floating-point interval.