Domain Of Hyperbolic Functions

Hyperbolic functions using osborn s rule which states that cos should be converted into cosh and sin into sinh except when there is a product of two sines when a sign change must be effected.

Domain of hyperbolic functions. The other hyperbolic functions are odd. Term by term differentiation yields differentiation formulas for the hyperbolic functions. Other related functions 9 1 c mathcentre january 9 2006.

The analogue of is. Definition 4 11 1 the hyperbolic cosine is the function cosh x e x e x over2 and the hyperbolic sine is the function sinh x e x e x over 2 notice that cosh is even that. Defining f x sinhx 4 4.

Most of the necessary range restrictions can be discerned by close examination of the graphs. The functions and sech x are defined for all real x. With appropriate range restrictions the hyperbolic functions all have inverses.

Defining f x coshx 2 3. These differentiation formulas give rise in turn to integration formulas. We have hyperbolic function identities like the trigonometric identities.

Since the area of a circular sector with radius r and angle u in radians is r 2 u 2 it will be equal to u when r 2 in the diagram such a circle is tangent to the hyperbola xy 1 at 1 1. The hyperbolic functions represent an expansion of trigonometry beyond the circular functions both types depend on an argument either circular angle or hyperbolic angle. Ify their domains define the reprocal functions sechx cschx and cothx.

The inverse hyperbolic functions are multiple valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single valued. The domains and ranges of the inverse hyperbolic functions are summarized in the following table. Their domains are all of except for the origin.