gives the hyperbolic cosecant of z.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- 1/Sinh[z] is automatically converted to Csch[z]. TrigFactorList[expr] does decomposition.
- For certain special arguments, Csch automatically evaluates to exact values.
- Csch can be evaluated to arbitrary numerical precision.
- Csch automatically threads over lists.
- Csch can be used with Interval and CenteredInterval objects. »
Background & Context
- Csch is the hyperbolic cosecant function, which is the hyperbolic analogue of the Csc circular function used throughout trigonometry. It is defined as the reciprocal of the hyperbolic sine function as . It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola . Csch[α] then represents the reciprocal of the vertical coordinate of the intersection point. Csch may also be defined as , where is the base of the natural logarithm Log.
- Csch automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Csch may be evaluated to arbitrary numeric precision. TrigFactorList can be used to factor expressions involving Csch into terms containing Sinh, Cosh, Sin, and Cos. Other operations useful for manipulation of symbolic expressions involving Csch include TrigToExp, TrigExpand, Simplify, and FullSimplify.
- Csch threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic cosecant of a square matrix (i.e. the power series for the hyperbolic cosecant function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic cosecants of the individual matrix elements.
- Csch[x] decreases exponentially as x approaches . Csch satisfies an identity similar to the Pythagorean identity satisfied by Csc, namely . The definition of the hyperbolic cosecant function is extended to complex arguments by way of the identity . Csch has poles at values for an integer and evaluates to ComplexInfinity at these points. Csch[z] has series expansion about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB.
- The inverse function of Csch is ArcCsch. Other related mathematical functions include Sinh and Sech.
Examplesopen allclose all
Basic Examples (5)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate Csch efficiently at high precision:
Csch threads elementwise over lists and matrices:
Csch can be used with Interval and CenteredInterval objects:
Specific Values (5)
Values of Csch at fixed purely imaginary points:
Singular point of Csch:
Find the value of satisfying equation :
Simple exact values are generated automatically:
More complicated cases require explicit use of FunctionExpand:
Plot the Csch function:
Function Properties (12)
Domain of Csch over reals:
Csch achieves all real values except 0:
Csch is an odd function:
Csch has the mirror property :
Csch is not an analytic function:
Csch is neither non-decreasing nor non-increasing:
Csch is injective:
Csch is not surjective:
Csch is neither non-negative nor non-positive:
Csch has both singularity and discontinuity at zero:
Csch is neither convex nor concave:
Indefinite integral of Csch:
Definite integral of an odd integrand over the interval centered at the origin is 0:
Series Expansions (3)
Integral Transforms (2)
Compute the Laplace transform using LaplaceTransform:
Function Identities and Simplifications (6)
Properties & Relations (9)
Basic parity and periodicity properties of Csch get automatically applied:
Use Simplify and FullSimplify to simplify expressions containing Csch:
Use FunctionExpand to express special values in radicals:
Compose with inverse functions:
Numerically find a root of a transcendental equation:
Obtain Csch from sums, products, and differential equations:
Csch appears in special cases of special functions:
Csch is a numeric function:
Possible Issues (5)
Machine-precision input is insufficient to give a correct answer:
Use arbitrary-precision evaluation instead:
A larger setting for $MaxExtraPrecision may be needed:
The inverse of Csch evaluates to Sinh:
No power series exists at infinity, where Csch has an essential singularity:
In TraditionalForm, parentheses are needed around the argument:
Neat Examples (1)
Plot Csch at infinity:
Wolfram Research (1988), Csch, Wolfram Language function, https://reference.wolfram.com/language/ref/Csch.html (updated 2021).
Wolfram Language. 1988. "Csch." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Csch.html.
Wolfram Language. (1988). Csch. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Csch.html