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Addition formulas exist in trigonometric functions. Double-angle andhalf-angle formulas can be derived from these formulas. Moreover, the relationequation between the trigonometric function and the hyperbolic function can bederived using an imaginary number. The inverse hyperbolic function$\mathrm{arsinh}(r)=\int_{0}^{r} \frac{1}{\sqrt{1+t^2} }\mathrm{d}t$ is similarto the inverse trigonometric function $\mathrm{arcsin}(r)=\int_{0}^{r}\frac{1}{\sqrt{1-t^2} }\mathrm{d}t$, such as the second degree of a polynomialand the constant term 1, except for the sign $-$ and $+$. Such an analogy holdsnot only when the degree of the polynomial is 2, but also for higher degrees.As such, a function exists with respect to the leaf function through theimaginary number $i$, such that the hyperbolic function exists with respect tothe trigonometric function through this imaginary number. In this study, werefer to this function as the hyperbolic leaf function. By making such adefinition, the relation equation between the leaf function and the hyperbolicleaf function makes it possible to easily derive various formulas, such asaddition formulas of hyperbolic leaf functions based on the addition formulasof leaf functions. Using the addition formulas, we can also derive the doubleangle and half-angle formulas. We then verify the consistency of these formulasby constructing graphs and numerical data.

Addition formulas of Leaf Functions and Hyperbolic Leaf Functions