# Sinh

The hyperbolic sine function $\mathrm{sinh} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by $$\mathrm{sinh}(z)=\dfrac{e^z-e^{-z}}{2}.$$ Since this function is one-to-one, its inverse function the inverse hyperbolic sine function is clear.

Domain coloring of $\sinh$.

# Properties

Derivative of sinh

Pythagorean identity for sinh and cosh

Relationship between sinh and hypergeometric 0F1

Weierstrass factorization of sinh

Taylor series for sinh

Relationship between Bessel I sub 1/2 and sinh

Relationship between sin and sinh

Relationship between sinh and sin

Relationship between tangent, Gudermannian, and sinh

Relationship between sinh, inverse Gudermannian, and tan

Period of sinh

Sum of cosh and sinh

Difference of cosh and sinh

Sinh is odd

Sinh of a sum

Cosh of a sum

Halving identity for sinh

Halving identity for tangent (2)

Halving identity for tangent (3)

Doubling identity for sinh (1)

Doubling identity for sinh (2)

Doubling identity for cosh (2)

Doubling identity for cosh (3)

# See Also

# References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.5.1$

**Hyperbolic trigonometric functions**