# Cosh

The hyperbolic cosine function $\cosh \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
$$\mathrm{cosh}(z)=\dfrac{e^z + e^{-z}}{2}.$$

Domain coloring of analytic continuation of $\cosh$.

# Properties[edit]

Derivative of cosh

Pythagorean identity for sinh and cosh

Weierstrass factorization of cosh

Relationship between cosh and hypergeometric 0F1

Relationship between Bessel I sub 1/2 and cosh

Relationship between cosh and cos

Relationship between cos and cosh

Relationship between secant, Gudermannian, and cosh

Relationship between cosh, inverse Gudermannian, and sec

Period of cosh

Sum of cosh and sinh

Difference of cosh and sinh

Cosh is even

Sinh of a sum

Cosh of a sum

Halving identity for sinh

Halving identity for cosh

Halving identity for tangent (1)

Halving identity for tangent (2)

Halving identity for tangent (3)

Doubling identity for sinh (1)

Doubling identity for cosh (1)

Doubling identity for cosh (2)

Doubling identity for cosh (3)

# See Also[edit]

# References[edit]

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.5.2$

**Hyperbolic trigonometric functions**