ArcCsch
ArcCsch[z]
gives the inverse hyperbolic cosecant 
of the complex number 
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcCsch automatically evaluates to exact values.
- ArcCsch can be evaluated to arbitrary numerical precision.
- ArcCsch automatically threads over lists.
- ArcCsch[z] has a branch cut discontinuity in the complex
plane running from 
to 
. - ArcCsch can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcCsch is the inverse hyperbolic cosecant function. For a real number
, ArcCsch[x] represents the hyperbolic angle measure 
such that 
. - ArcCsch automatically threads over lists. For certain special arguments, ArcCsch automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCsch may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCsch include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCsch is defined for complex argument
by 
. ArcCsch[z] has a branch cut discontinuity in the complex 
plane. - Related mathematical functions include Csch, ArcSech, and ArcCsc.
Examples
open all close allBasic Examples (6)
ArcCsch[4.5]Plot over a subset of the reals:
Plot[ArcCsch[x], {x, -5, 5}]Plot over a subset of the complexes:
ComplexPlot3D[ArcCsch[z], {z, -3 - 4I, 3 + 4I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[ArcCsch[x], {x, 0, 10}, Assumptions -> x > 0]Asymptotic expansion at Infinity:
Series[ArcCsch[x], {x, ∞, 5}]Asymptotic expansion at a singular point:
Series[ArcCsch[x], {x, I, 3}, Assumptions -> x > 0]Scope (43)
Numerical Evaluation (6)
ArcCsch[5.0]N[ArcCsch[5], 50]The precision of the output tracks the precision of the input:
ArcCsch[5.000000000000000000000000000000000000000000]Evaluate for complex arguments:
ArcCsch[1.5 + I]Evaluate ArcCsch efficiently at high precision:
ArcCsch[5`500]//TimingArcCsch[5`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcCsch[Interval[{-4, Csch[2]}]]ArcCsch[CenteredInterval[3, 1 / 100]]ArcCsch[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcCsch[Around[.9, 0.1]]Compute the elementwise values of an array:
ArcCsch[{{2I, 0}, {I, -2I}}]Or compute the matrix ArcCsch function using MatrixFunction:
MatrixFunction[ArcCsch[#]&, {{2I, 0}, {I, -2I}}]Specific Values (4)
Values of ArcCsch at fixed points:
Table[ArcCsch[I n], {n, -2, 2}]ArcCsch[Infinity]ArcCsch[ComplexInfinity]Singular point of ArcCsch:
ArcCsch[0]Find the value of 
satisfying equation 
:
f[x_] := ArcCsch[x] - Log[2];sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, 0, 2}, Epilog -> Style[Point[{xzero, f[xzero]}], PointSize[Large], Red]]Visualization (3)
Plot the ArcCsch function:
Plot[ArcCsch[x], {x, -1, 1}]
ComplexContourPlot[Re[ArcCsch[z]], {z, -3 - 3 I, 3 + 3 I}, Contours -> 24]
ComplexContourPlot[Im[ArcCsch[z]], {z, -3 - 3 I, 3 + 3 I}, Contours -> 24]
Table[PolarPlot[ArcCsch[k ϕ], {ϕ, -3π, 3π}, Sequence[Frame -> True, PlotLabel -> "k=" <> ToString[k], PlotRange -> {{-6, 6}, {-0.3, 0.8}}, AspectRatio -> 0.8]], {k, 1, 4}]Function Properties (11)
ArcCsch is defined for all real values except 0:
FunctionDomain[ArcCsch[x], x]FunctionDomain[ArcCsch[z], z, Complexes]ArcCsch achieves all real values except 0:
FunctionRange[ArcCsch[x], x, y]Function range for arguments from the complex domain:
FunctionRange[ArcCsch[z], z, y, Complexes]ArcCsch is an odd function:
ArcCsch[-x]ArcCsch has the mirror property ![csch^(-1)(TemplateBox[{z}, Conjugate])=TemplateBox[{{{csch, ^, {(, {-, 1}, )}}, (, z, )}}, Conjugate]](Files/ArcCsch.en/17.png "csch^(-1)(TemplateBox[{z}, Conjugate])=TemplateBox[{{{csch, ^, {(, {-, 1}, )}}, (, z, )}}, Conjugate]"){kind=small}
:
FullSimplify[ArcCsch[Conjugate[z]] == Conjugate[ArcCsch[z]]]ArcCsch is not an analytic function:
FunctionAnalytic[ArcCsch[x], x]FunctionMeromorphic[ArcCsch[x], x]ArcCsch is neither non-decreasing nor non-increasing:
FunctionMonotonicity[ArcCsch[x], x]ArcCsch is injective:
FunctionInjective[ArcCsch[x], x]Plot[{ArcCsch[x], 1}, {x, -3, 3}]FunctionSurjective[ArcCsch[x], x]Limit[ArcCsch[x], x -> ∞]Plot[{ArcCsch[x], 0}, {x, -1, 1}]ArcCsch is neither non-negative nor non-positive:
FunctionSign[ArcCsch[x], x]It has both singularity and discontinuity at zero:
FunctionSingularities[ArcCsch[x], x]FunctionDiscontinuities[ArcCsch[x], x]ArcCsch is neither convex nor concave:
FunctionConvexity[ArcCsch[x], x]TraditionalForm formatting:
ArcCsch[x]//TraditionalFormDifferentiation (3)
D[ArcCsch[x], x]Table[D[ArcCsch[x], {x, n}], {n, 1, 4}]//FullSimplifyPlot[Evaluate[%], {x, -1, 1}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]
derivative:D[ArcCsch[x], {x, n}]Integration (3)
Indefinite integral of ArcCsch:
Integrate[ArcCsch[x], x]Definite integral of ArcCsch over the interval centered at the origin is 0:
Integrate[ArcCsch[x], {x, -1, 1}, PrincipalValue -> True]Integrate[x ArcCsch[x], x]Integrate[x ^ k ArcCsch[x], x]Series Expansions (3)
Find the Taylor expansion using Series:
Series[ArcCsch[x], {x, 1, 4}]Plot the first three approximations for ArcCsch around 
:
terms = Normal@Table[Series[ArcCsch[x], {x, 1, m}], {m, 1, 3}];
Plot[{ArcCsch[x], terms}, {x, 0, 2}, PlotRange -> {{0, 2}, {0, 5}}]Find series expansions at branch points and branch cuts:
Series[ArcCsch[x], {x, 0, 1}]Series[ArcCsch[x], {x, (I/2), 1}]Apply ArcCsch to a power series:
ArcCsch[Exp[x] + O[x] ^ 6]Integral Transforms (2)
Compute the Mellin transform using MellinTransform:
MellinTransform[ArcCsch[x], x, s ]HankelTransform[ArcCsch[r], r, s]Function Identities and Simplifications (3)
Simplification of ArcCsch:
x^4 y^4 - 2 x^2 y^2 (2 + x^2 + y^2) Csch[ArcCsch[x] + ArcCsch[y]]^2 + (x^2 - y^2)^2 Csch[ArcCsch[x] + ArcCsch[y]]^4//FullSimplifyUse TrigToExp to express in terms of logarithm:
ArcCsch[z]//TrigToExpExpToTrig[%]//FullSimplifyExpand assuming real variables 
and 
:
ComplexExpand[ArcCsch[x + I y]]Function Representations (5)
Represent using ArcSinh:
ArcSinh[1 / x]//FullSimplifyRepresentation through inverse Jacobi functions:
InverseJacobiCS[z, 1]InverseJacobiDS[z, 1]Represent using Hypergeometric2F1:
(HypergeometricPFQ[{(1/2), (1/2)}, {(3/2)}, -(1/z^2)]/z)//FullSimplifyArcCsch can be represented in terms of MeijerG:
MeijerGReduce[ArcCsch[x], x]Activate[%]//FullSimplifyArcCsch can be represented as a DifferentialRoot:
DifferentialRootReduce[ArcCsch[x], x]Applications (3)
Branch cut structure of ArcCsch runs along the imaginary axis:
Plot3D[Re[ArcCsch[x - I y]], {y, -2, 2}, {x, -2, 2}]Solve a differential equation:
DSolve[x y'[x] - y[x] - ArcCsch[x] == 0, y[x], x]Solution of the sinh‐Gordon equation 
:
u[x_, t_] := 2ArcCsch[JacobiSC[x - t, m]]FullSimplify[D[u[x, t], x, t] + Sinh[u[x, t]]]Plot3D[Evaluate[u[x, t] /. m -> 4 / 3], {x, -3, 2}, {t, 0, 5 / 2}, MaxRecursion -> 1]Properties & Relations (2)
Compose with the inverse function:
{ArcCsch[Csch[z]], Csch[ArcCsch[z]]}Use PowerExpand to disregard multivaluedness of the ArcCsch:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcCsch[Csch[z]], z∈Reals]Use TrigToExp to express in terms of logarithm:
ArcCsch[z]//TrigToExpExpToTrig[%]Related Links
MathWorld
The Wolfram Functions SiteText
Wolfram Research (1988), ArcCsch, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCsch.html (updated 2021).
CMS
Wolfram Language. 1988. "ArcCsch." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCsch.html.
APA
Wolfram Language. (1988). ArcCsch. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCsch.html