Domain Hyperbolic Functions

Hyperbolic functions are defined in terms of exponential functions.

Domain hyperbolic functions. As the ratio of the hyperbolic sine and cosine functions that are particular cases of the generalized hypergeometric bessel struve and mathieu functions the hyperbolic tangent function can also be represented as ratios of those special functions. The function satisfies the conditions cosh0 1 and coshx cosh x. If x sinh y then y sinh 1 a is called the inverse hyperbolic sine of x.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. The hyperbolic function f x coshx is defined by the formula coshx ex e x 2. The effects of a and q on f x frac a x q.

Functions of the general form y frac a x q are called hyperbolic functions where a and q are constants. They also occur in the. In mathematics hyperbolic functions are analogues of the ordinary trigonometric functions but defined using the hyperbola rather than the circle.

These differentiation formulas give rise in turn to integration formulas. The hyperbolic sine and hyperbolic cosine are defined by the hyperbolic tangent and hyperbolic cotangent are defined by the hyperbolic sine. Similarly we define the other inverse hyperbolic functions.

Consideration of hyperbolic functions was done by the swiss mathematician johann heinrich lambert 1728 1777. The function is continuous on its domain unbounded and symmetric namely odd since we have sinh x sinh x. The effect of q on vertical shift.

The graph of coshx is always above the graphs of ex 2 and e x 2. 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. 2 1 definitions the hyperbolic cosine function written cosh x is defined for all real values of x by the relation cosh x 1 2 ex e x similarly the hyperbolic sine function sinh x is defined by sinh x 1 2 ex e x the.