ArcCsc
ArcCsc[z]
gives the arc cosecant 
of the complex number 
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- All results are given in radians.
- For real z outside the interval
to 
, the results are always in the range 
to 
, excluding 0. - For certain special arguments, ArcCsc automatically evaluates to exact values.
- ArcCsc can be evaluated to arbitrary numerical precision.
- ArcCsc automatically threads over lists.
- ArcCsc[z] has a branch cut discontinuity in the complex
plane running from 
to 
. - ArcCsc can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcCsc is the inverse cosecant function. For a real number
, ArcCsc[x] represents the radian angle measure 
, 
, such that 
. - ArcCsc automatically threads over lists. For certain special arguments, ArcCsc automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCsc may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCsc include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCsc is defined for complex argument
via 
. ArcCsc[z] has a branch cut discontinuity in the complex 
plane. - Related mathematical functions include Csc, ArcSec, and ArcCsch.
Examples
open all close allBasic Examples (5)
ArcCsc[1]Divide by Degree to get results in degrees:
ArcCsc[1] / Degree//FullSimplifyPlot over a subset of the reals:
Plot[ArcCsc[x], {x, -5, 5}]Plot over a subset of the complexes:
ComplexPlot3D[ArcCsc[z], {z, -4 - 2I, 4 + 2I}, PlotLegends -> Automatic]Asymptotic expansion at Infinity:
Series[ArcCsc[x], {x, ∞, 5}]Asymptotic expansion at a singular point:
Series[ArcCsc[x], {x, 1, 3}, Assumptions -> x > 1]//FullSimplifyScope (41)
Numerical Evaluation (6)
ArcCsc[5.0]N[ArcCsc[5], 50]The precision of the output tracks the precision of the input:
ArcCsc[5.00000000000000000000000000000000000000000]Evaluate for complex arguments:
ArcCsc[0.5 + 2.3I]Evaluate ArcCsc efficiently at high precision:
ArcCsc[5`500]//TimingArcCsc[5`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcCsc[Interval[{1, 5}]]ArcCsc[CenteredInterval[-3, 1 / 100]]ArcCsc[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcCsc[Around[(2/Sqrt[3]), 0.1]]Compute the elementwise values of an array:
ArcCsc[{{Sqrt[2], -Sqrt[2]}, {0, -Sqrt[2]}}]Or compute the matrix ArcCsc function using MatrixFunction:
MatrixFunction[ArcCsc[#]&, {{Sqrt[2], -Sqrt[2]}, {0, -Sqrt[2]}}]Specific Values (4)
Values of ArcCsc at fixed points:
{ArcCsc[-Sqrt[2]], ArcCsc[-(2/Sqrt[3])], ArcCsc[(2/Sqrt[3])], ArcCsc[Sqrt[2]]}ArcCsc[Infinity]ArcCsc[ComplexInfinity]ArcCsc[{1, -1, 0}]ArcCsc is not differentiable at these points:
ArcCsc'[{1, -1, 0}]//QuietFind the value of 
satisfying equation 
:
f[x_] := ArcCsc[x] - (π/3);sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, 1, 2}, Epilog -> Style[Point[{xzero, f[xzero]}], PointSize[Large], Red]]Visualization (3)
Plot the ArcCsc function:
Plot[ArcCsc[x], {x, -2, 2}]
ComplexContourPlot[Re[ArcCsc[z]], {z, -2π - 2π I, 2π + 2π I}, IconizedObject[«PlotOptions»]]
ComplexContourPlot[Im[ArcCsc[z]], {z, -2π - 2π I, 2π + 2π I}, IconizedObject[«PlotOptions»]]
Table[PolarPlot[ArcCsc[k ϕ], {ϕ, -4π, 4π}, Sequence[Frame -> True, PlotLabel -> "k=" <> ToString[k], FrameTicks -> {{{-1, 0, 1}, None}, {{-1, 0, 1}, None}}, PlotRange -> {{-1, 1}, {-1, 1}}]], {k, 1, 4}]Function Properties (11)
ArcCsc is defined for all real values except from the interval 
:
FunctionDomain[ArcCsc[x], x]FunctionDomain[ArcCsc[z], z, Complexes]ArcCsc achieves all real values from the interval 
except 0:
FunctionRange[ArcCsc[x], x, y]Function range for arguments from the complex domain:
FunctionRange[ArcCsc[z], z, y, Complexes]ArcCsc is an odd function:
ArcCsc[-x]ArcCsc is not an analytic function:
FunctionAnalytic[ArcCsc[x], x]FunctionMeromorphic[ArcCsc[x], x]ArcCsc is monotonic in a specific range:
FunctionMonotonicity[ArcCsc[x], x]FunctionMonotonicity[{ArcCsc[x], x ≤ -1 }, x]FunctionMonotonicity[{ArcCsc[x], x ≥ 1}, x]Plot[ArcCsc[x], {x, -3, 3}]ArcCsc is injective:
FunctionInjective[ArcCsc[x], x]Plot[{ArcCsc[x], 1 / 2}, {x, -5, 5}]ArcCsc is not surjective:
FunctionSurjective[ArcCsc[x], x]Plot[{ArcCsc[x], 2}, {x, -5, 5}]ArcCsc is neither non-negative nor non-positive:
FunctionSign[ArcCsc[x], x]It has both singularity and discontinuity for x in [-1,1]:
FunctionSingularities[ArcCsc[x], x]FunctionDiscontinuities[ArcCsc[x], x]ArcCsc is neither convex nor concave:
FunctionConvexity[ArcCsc[x], x]TraditionalForm formatting:
ArcCsc[x]//TraditionalFormDifferentiation (3)
D[ArcCsc[x], x]Table[D[ArcCsc[x], {x, n}], {n, 1, 4}]//FullSimplifyPlot[Evaluate[%], {x, -3, 3}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]
derivative:D[ArcCsc[x], {x, n}]Integration (3)
Series Expansions (3)
Find the Taylor expansion using Series:
Series[ArcCsc[x], {x, 2, 4}]Plots of the first three approximations for ArcCsc around 
:
terms = Normal@Table[Series[ArcCsc[x], {x, 2, m}], {m, 1, 3}];
Plot[{ArcCsc[x], terms}, {x, 1, 3}, PlotRange -> {{1, 3}, {0, 3}}]Find series expansions at branch points and branch cuts:
Series[ArcCsc[x], {x, 1, 1}]Series[ArcCsc[x], {x, -1 / 2, 1}]ArcCsc can be applied to power series:
ArcCsc[x + x ^ 2 + O[x, Infinity] ^ 4]Function Identities and Simplifications (3)
Simplify expressions involving ArcCsc:
FullSimplify[ArcCsc[Sqrt[(1/1 - x^2)]], 0 ≤ x ≤ 1]Compose with inverse functions:
{ArcCsc[Csc[z]], Csc[ArcCsc[z]]}Use PowerExpand to disregard multivaluedness of the ArcCsc:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcCsc[Csc[z]], 0 < z < Pi / 2]Use TrigToExp to express in terms of logarithm:
ArcCsc[z]//TrigToExpUse ExpToTrig to convert back:
ExpToTrig[%]//FullSimplifyFunction Representations (5)
Represent using ArcSin:
ArcSin[(1/z)]//FullSimplifyRepresentation through inverse Jacobi functions:
InverseJacobiDS[z, 0]InverseJacobiNS[z, 0]Represent using Hypergeometric2F1:
(Hypergeometric2F1[(1/2), (1/2), (3/2), (1/z^2)]/z)//FullSimplifyArcCsc can be represented in terms of MeijerG:
MeijerGReduce[ArcCsc[x], x]Activate[%]//FullSimplifyArcCsc can be represented as a DifferentialRoot:
DifferentialRootReduce[ArcCsc[x], x]Applications (3)
Branch cut of ArcCsc runs along the real axis:
Plot3D[Im[ArcCsc[x - I y]], {x, -2, 2}, {y, -2, 2}]Solve a differential equation:
DSolve[2 x Sin[(y[x]/x)] - Cot[(y[x]/x)] y[x] + x Cot[(y[x]/x)] y'[x] == 0, y[x], x]//QuietDistribution of the average distance s of all pairs of points in a d‐dimensional hypersphere:
p[d_, s_] := d s ^ (d - 1) BetaRegularized[1 - s ^ 2 / 4, (d + 1) / 2, 1 / 2]Low‐dimensional distributions can be expressed in elementary functions:
Table[FullSimplify[FunctionExpand[p[d, s]], s > 0], {d, 3}]Plot[Evaluate[Table[p[d, s], {d, 10}]], {s, 0, 2}]Properties & Relations (5)
Compose with inverse functions:
{ArcCsc[Csc[z]], Csc[ArcCsc[z]]}Use PowerExpand to disregard multivaluedness of the ArcCsc:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcCsc[Csc[z]], 0 < z < Pi / 2]Use TrigToExp to express in terms of logarithm:
ArcCsc[z]//TrigToExpUse ExpToTrig to convert back:
ExpToTrig[%]ArcCsc gives the angle in radians, while ArcCscDegrees gives the same angle in degrees:
ArcCsc[2]ArcCscDegrees[2]Use FunctionExpand to convert trigs of arctrigs into an algebraic function:
FunctionExpand[Tan[5ArcCsc[z]]]Simplify[%]Use Reduce to solve equations involving ArcCsc:
Reduce[ArcCsc[z] + ArcCsc[z + 1] == Pi / 2, z]//QuietRelated Links
MathWorld
The Wolfram Functions SiteText
Wolfram Research (1988), ArcCsc, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCsc.html (updated 2021).
CMS
Wolfram Language. 1988. "ArcCsc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCsc.html.
APA
Wolfram Language. (1988). ArcCsc. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCsc.html