Sinh
✖
Sinh
Details
- 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.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Scope (47)Survey of the scope of standard use cases
Numerical Evaluation (6)
https://wolfram.com/xid/0mq4vc-l274ju
https://wolfram.com/xid/0mq4vc-0e
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0mq4vc-gxh
Sinh can take complex number inputs:
https://wolfram.com/xid/0mq4vc-bgv
Evaluate Sinh efficiently at high precision:
https://wolfram.com/xid/0mq4vc-di5gcr
https://wolfram.com/xid/0mq4vc-bq2c6r
Compute the elementwise values of an array using automatic threading:
https://wolfram.com/xid/0mq4vc-thgd2
Or compute the matrix Sinh function using MatrixFunction:
https://wolfram.com/xid/0mq4vc-o5jpo
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
https://wolfram.com/xid/0mq4vc-c2ol
https://wolfram.com/xid/0mq4vc-lmyeh7
https://wolfram.com/xid/0mq4vc-j0p7jj
Or compute average-case statistical intervals using Around:
https://wolfram.com/xid/0mq4vc-cw18bq
Specific Values (4)
Values of Sinh at fixed, purely imaginary points:
https://wolfram.com/xid/0mq4vc-nww7l
https://wolfram.com/xid/0mq4vc-8rwwz
https://wolfram.com/xid/0mq4vc-d5y0fe
Zero of Sinh:
https://wolfram.com/xid/0mq4vc-cw39qs
Find the zero of Sinh using Solve:
https://wolfram.com/xid/0mq4vc-bgtr3o
https://wolfram.com/xid/0mq4vc-oiazhe
https://wolfram.com/xid/0mq4vc-bordz7
Simple exact values are generated automatically:
https://wolfram.com/xid/0mq4vc-eib
More complicated cases require explicit use of FunctionExpand:
https://wolfram.com/xid/0mq4vc-mnn
https://wolfram.com/xid/0mq4vc-p3j
Visualization (3)
Plot the Sinh function:
https://wolfram.com/xid/0mq4vc-ecj8m7
https://wolfram.com/xid/0mq4vc-bo5grg
https://wolfram.com/xid/0mq4vc-bjwtin
https://wolfram.com/xid/0mq4vc-epb4bn
Function Properties (12)
Sinh is defined for all real and complex values:
https://wolfram.com/xid/0mq4vc-cl7ele
https://wolfram.com/xid/0mq4vc-de3irc
Sinh achieves all real values:
https://wolfram.com/xid/0mq4vc-evf2yr
The range for complex values is the whole plane:
https://wolfram.com/xid/0mq4vc-fphbrc
Sinh is an odd function:
https://wolfram.com/xid/0mq4vc-dnla5q
Sinh has the mirror property :
https://wolfram.com/xid/0mq4vc-heoddu
Sinh is an analytic function of x:
https://wolfram.com/xid/0mq4vc-h5x4l2
Sinh is monotonic:
https://wolfram.com/xid/0mq4vc-g6kynf
Sinh is injective:
https://wolfram.com/xid/0mq4vc-gi38d7
https://wolfram.com/xid/0mq4vc-ctca0g
Sinh is surjective:
https://wolfram.com/xid/0mq4vc-hkqec4
https://wolfram.com/xid/0mq4vc-hdm869
Sinh is neither non-negative nor non-positive:
https://wolfram.com/xid/0mq4vc-84dui
Sinh has no singularities or discontinuities:
https://wolfram.com/xid/0mq4vc-mdtl3h
https://wolfram.com/xid/0mq4vc-mn5jws
Sinh is neither convex nor concave:
https://wolfram.com/xid/0mq4vc-io426y
TraditionalForm formatting:
https://wolfram.com/xid/0mq4vc-ba2kou
Differentiation (3)
Integration (3)
Indefinite integral of Sinh:
https://wolfram.com/xid/0mq4vc-bponid
Definite integral of an odd integrand over the interval centered at the origin is 0:
https://wolfram.com/xid/0mq4vc-b9jw7l
https://wolfram.com/xid/0mq4vc-cwqkze
https://wolfram.com/xid/0mq4vc-fo8r1r
Series Expansions (4)
Find the Taylor expansion using Series:
https://wolfram.com/xid/0mq4vc-ewr1h8
Plot the first three approximations for Sinh around :
https://wolfram.com/xid/0mq4vc-binhar
General term in the series expansion of Sinh:
https://wolfram.com/xid/0mq4vc-dznx2j
A few first terms of Fourier series:
https://wolfram.com/xid/0mq4vc-f64drv
Sinh can be applied to power series:
https://wolfram.com/xid/0mq4vc-d486tg
Integral Transforms (2)
Compute the Laplace transform using LaplaceTransform:
https://wolfram.com/xid/0mq4vc-fc7wyp
https://wolfram.com/xid/0mq4vc-c4435i
https://wolfram.com/xid/0mq4vc-daxi8b
Function Identities and Simplifications (6)
Sinh of a double angle:
https://wolfram.com/xid/0mq4vc-mjplp7
Sinh of a sum:
https://wolfram.com/xid/0mq4vc-nfe4y
Convert multiple‐angle expressions:
https://wolfram.com/xid/0mq4vc-d72
https://wolfram.com/xid/0mq4vc-jzs
Convert sums of hyperbolic functions to products:
https://wolfram.com/xid/0mq4vc-fwl
Expand assuming real variables and :
https://wolfram.com/xid/0mq4vc-lqhbu
https://wolfram.com/xid/0mq4vc-u1m
Function Representations (4)
Representation through Sin:
https://wolfram.com/xid/0mq4vc-df304y
Representation through Bessel functions:
https://wolfram.com/xid/0mq4vc-ewa69v
https://wolfram.com/xid/0mq4vc-9y9s4
Representation in terms of MeijerG:
https://wolfram.com/xid/0mq4vc-ekhdli
https://wolfram.com/xid/0mq4vc-bpnrbj
Sinh can be represented as a DifferentialRoot:
https://wolfram.com/xid/0mq4vc-bgjnbg
Applications (8)Sample problems that can be solved with this function
https://wolfram.com/xid/0mq4vc-rjj
Rotation matrix in hyperbolic space:
https://wolfram.com/xid/0mq4vc-sxe
Build from infinitesimal transformations:
https://wolfram.com/xid/0mq4vc-o83
https://wolfram.com/xid/0mq4vc-piw
https://wolfram.com/xid/0mq4vc-w85
The matrix is orthogonal with respect to the Minkowski metric:
https://wolfram.com/xid/0mq4vc-wue
Construct a relativistic coordinate transformation for rapidity :
https://wolfram.com/xid/0mq4vc-v34
https://wolfram.com/xid/0mq4vc-egu
Prolate spheroidal coordinates:
https://wolfram.com/xid/0mq4vc-3
Special solution of the sine–Gordon equation:
https://wolfram.com/xid/0mq4vc-juj
https://wolfram.com/xid/0mq4vc-kh9
Solve a differential equation:
https://wolfram.com/xid/0mq4vc-dat1tz
Compute the arc length of a hyperbola as a function of the angle of a point on the hyperbola with Sinh and Cosh:
https://wolfram.com/xid/0mq4vc-cff5ot
https://wolfram.com/xid/0mq4vc-bglni
Plot the arc length as a function of the angle:
https://wolfram.com/xid/0mq4vc-ih7y0
Find a point in the hyperbola using Cosh and Sinh functions:
https://wolfram.com/xid/0mq4vc-u80xp
Properties & Relations (11)Properties of the function, and connections to other functions
Basic parity and periodicity properties of the hyperbolic sine function get automatically applied:
https://wolfram.com/xid/0mq4vc-22
https://wolfram.com/xid/0mq4vc-h2c
https://wolfram.com/xid/0mq4vc-t93
Complicated expressions containing hyperbolic functions do not autosimplify:
https://wolfram.com/xid/0mq4vc-eqs
https://wolfram.com/xid/0mq4vc-ny
Compose with inverse functions:
https://wolfram.com/xid/0mq4vc-e2j
https://wolfram.com/xid/0mq4vc-kmb
https://wolfram.com/xid/0mq4vc-med
Numerically find a root of a transcendental equation:
https://wolfram.com/xid/0mq4vc-njc
https://wolfram.com/xid/0mq4vc-lc0
https://wolfram.com/xid/0mq4vc-myq
https://wolfram.com/xid/0mq4vc-eluutt
Sinh appears in special cases of many mathematical functions:
https://wolfram.com/xid/0mq4vc-t75
Sinh is a numeric function:
https://wolfram.com/xid/0mq4vc-jik
https://wolfram.com/xid/0mq4vc-qx1
The generating function for Sinh:
https://wolfram.com/xid/0mq4vc-pz93yz
The exponential generating function for Sinh:
https://wolfram.com/xid/0mq4vc-gaiyeu
Possible Issues (5)Common pitfalls and unexpected behavior
Machine-precision input is insufficient to give a correct answer:
https://wolfram.com/xid/0mq4vc-za
With exact input, the answer is correct:
https://wolfram.com/xid/0mq4vc-bfw
A larger setting for $MaxExtraPrecision can be needed:
https://wolfram.com/xid/0mq4vc-pik
https://wolfram.com/xid/0mq4vc-uz5
Machine number inputs can give high‐precision results:
https://wolfram.com/xid/0mq4vc-hr
https://wolfram.com/xid/0mq4vc-i6f
No power series exists at infinity, where Sinh has an essential singularity:
https://wolfram.com/xid/0mq4vc-les
In TraditionalForm, parentheses are needed around the argument:
https://wolfram.com/xid/0mq4vc-sfb
https://wolfram.com/xid/0mq4vc-25
Wolfram Research (1988), Sinh, Wolfram Language function, https://reference.wolfram.com/language/ref/Sinh.html (updated 2021).
Text
Wolfram Research (1988), Sinh, Wolfram Language function, https://reference.wolfram.com/language/ref/Sinh.html (updated 2021).
Wolfram Research (1988), Sinh, Wolfram Language function, https://reference.wolfram.com/language/ref/Sinh.html (updated 2021).
CMS
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. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sinh.html.
APA
Wolfram Language. (1988). Sinh. Wolfram Language & System Documentation Center. Retrieved from 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
BibTeX
@misc{reference.wolfram_2024_sinh, author="Wolfram Research", title="{Sinh}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sinh.html}", note=[Accessed: 07-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_sinh, organization={Wolfram Research}, title={Sinh}, year={2021}, url={https://reference.wolfram.com/language/ref/Sinh.html}, note=[Accessed: 07-January-2025
]}