gives the hyperbolic cosine of z.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, Cosh automatically evaluates to exact values.
- Cosh can be evaluated to arbitrary numerical precision.
- Cosh automatically threads over lists.
- Cosh can be used with Interval and CenteredInterval objects. »
Background & Context
- Cosh is the hyperbolic cosine function, which is the hyperbolic analogue of the Cos 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 . Cosh[α] then represents the horizontal coordinate of the intersection point. Cosh may also be defined as , where is the base of the natural logarithm Log.
- Cosh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numerical expressions as arguments, Cosh may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Cosh include TrigToExp, TrigExpand, Simplify, and FullSimplify.
- Cosh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic cosine of a square matrix (i.e. the power series for the hyperbolic cosine function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic cosines of the individual matrix elements.
- Cosh[x] increases exponentially as x approaches . Cosh satisfies an identity similar to the Pythagorean identity satisfied by Cos, namely . The definition of the hyperbolic sine function is extended to complex arguments by way of the identity . The hyperbolic cosine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Cosh[z] has series expansion about the origin.
- The inverse function of Cosh is ArcCosh. Additional related mathematical functions include Sinh and Sech.
Examplesopen allclose all
Basic Examples (4)
Numerical Evaluation (6)
Cosh can take complex number inputs:
Evaluate Cosh efficiently at high precision:
Cosh threads elementwise over lists and matrices:
Specific Values (4)
Plot the Cosh function:
Function Properties (12)
Cosh is defined for all real and complex values:
Cosh achieves all real values greater than or equal to 1:
Cosh is an even function:
Cosh has the mirror property :
Cosh is an analytic function of x:
Cosh is neither non-decreasing nor non-increasing:
Cosh is not injective:
Cosh is not surjective:
Cosh is non-negative:
Cosh is convex:
Series Expansions (4)
Function Identities and Simplifications (6)
Properties & Relations (12)
Basic parity and periodicity properties of Cosh get automatically applied:
Cosh appears in special cases of many mathematical functions:
Cosh is a numeric function:
The generating function for Cosh:
The exponential generating function for Cosh:
Possible Issues (5)
A larger setting for $MaxExtraPrecision can be needed:
No power series exists at infinity, where Cosh has an essential singularity:
In TraditionalForm, parentheses are needed around the argument:
Wolfram Research (1988), Cosh, Wolfram Language function, https://reference.wolfram.com/language/ref/Cosh.html (updated 2021).
Wolfram Language. 1988. "Cosh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Cosh.html.
Wolfram Language. (1988). Cosh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Cosh.html