gives the hyperbolic sine of z.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, Sinh automatically evaluates to exact values.
- Sinh can be evaluated to arbitrary numerical precision.
- Sinh automatically threads over lists.
- Sinh can be used with Interval and CenteredInterval objects. »
Background & Context
- Sinh is the hyperbolic sine function, which is the hyperbolic analogue of the Sin circular function used throughout trigonometry. 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 . Sinh[α] then gives the vertical coordinate of the intersection point. Sinh may also be defined as , where is the base of the natural logarithm Log.
- Sinh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Sinh may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Sinh include TrigToExp, TrigExpand, Simplify, and FullSimplify.
- Sinh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic sine of a square matrix (i.e. the power series for the hyperbolic sine function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic sines of the individual matrix elements.
- Sinh[x] decreases exponentially as x approaches and increases exponentially as x approaches . Sinh satisfies an identity similar to the Pythagorean identity satisfied by Sin, namely . The definition of the hyperbolic sine function is extended to complex arguments by way of the identity . The hyperbolic sine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Sinh[z] has series expansion about the origin.
- The inverse function of Sinh is ArcSinh. Related mathematical functions include Cosh and Csch.
Examplesopen allclose all
Basic Examples (4)
Numerical Evaluation (6)
Sinh can take complex number inputs:
Evaluate Sinh efficiently at high precision:
Sinh threads elementwise over lists and matrices:
Specific Values (4)
Plot the Sinh function:
Function Properties (12)
Sinh is defined for all real and complex values:
Sinh achieves all real values:
Sinh is an odd function:
Sinh has the mirror property :
Sinh is an analytic function of x:
Sinh is monotonic:
Sinh is injective:
Sinh is surjective:
Sinh is neither non-negative nor non-positive:
Sinh has no singularities or discontinuities:
Sinh is neither convex nor concave:
Indefinite integral of Sinh:
Series Expansions (4)
Function Identities and Simplifications (6)
Properties & Relations (11)
Sinh appears in special cases of many mathematical functions:
Sinh is a numeric function:
The generating function for Sinh:
The exponential generating function for Sinh:
Possible Issues (5)
A larger setting for $MaxExtraPrecision can be needed:
No power series exists at infinity, where Sinh has an essential singularity:
In TraditionalForm, parentheses are needed around the argument:
Wolfram Research (1988), Sinh, Wolfram Language function, https://reference.wolfram.com/language/ref/Sinh.html (updated 2021).
Wolfram Language. 1988. "Sinh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sinh.html.
Wolfram Language. (1988). Sinh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sinh.html