# Cosh

Cosh[z]

gives the hyperbolic cosine of z.

# Details

• 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.

# 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.

# Examples

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## Basic Examples(3)

Evaluate numerically:

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Plot over a subset of the reals:

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Series expansion:

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