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)
The precision of the output tracks the precision of the input:
Sinh can take complex number inputs:
Evaluate Sinh efficiently at high precision:
Sinh threads elementwise over lists and matrices:
Sinh can be used with Interval and CenteredInterval objects:
Specific Values (4)
Values of Sinh at fixed, purely imaginary points:
Zero of Sinh:
Find the zero of Sinh using Solve:
Simple exact values are generated automatically:
More complicated cases require explicit use of FunctionExpand:
Plot the Sinh function:
Function Properties (12)
Sinh is defined for all real and complex values:
Sinh achieves all real values:
The range for complex values is the whole plane:
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:
Definite integral of an odd integrand over the interval centered at the origin is 0:
Series Expansions (4)
Integral Transforms (2)
Compute the Laplace transform using LaplaceTransform:
Function Identities and Simplifications (6)
Function Representations (4)
Representation through Sin:
Representation through Bessel functions:
Representation in terms of MeijerG:
Sinh can be represented as a DifferentialRoot:
Rotation matrix in hyperbolic space:
Build from infinitesimal transformations:
The matrix is orthogonal with respect to the Minkowski metric:
Construct a relativistic coordinate transformation for rapidity :
Prolate spheroidal coordinates:
Special solution of the sine–Gordon equation:
Solve a differential equation:
Compute the arc length of a hyperbola as a function of the angle of a point on the hyperbola with Sinh and Cosh:
Plot the arc length as a function of the angle:
Find a point in the hyperbola using Cosh and Sinh functions:
Properties & Relations (11)
Basic parity and periodicity properties of the hyperbolic sine function get automatically applied:
Complicated expressions containing hyperbolic functions do not autosimplify:
Compose with inverse functions:
Numerically find a root of a transcendental equation:
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)
Machine-precision input is insufficient to give a correct answer:
With exact input, the answer is correct:
A larger setting for $MaxExtraPrecision can be needed:
Machine number inputs can give high‐precision results:
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