Hyperbolic function

The hyperbolic functions are:

  • Hyperbolic sine ( abbreviated sinh )
  • Hyperbolic cosine ( cosh )
  • Hyperbolic tangent ( tanh )
  • Hyperbolic cotangent ( coth )
  • Hyperbolic secant ( sech )
  • Hyperbolic cosecant ( csch ).

Sinh and cosh are defined for all complex numbers and holomorphic in the entire field of complex numbers. The other hyperbolic functions have poles on the imaginary axis.

  • 3.1 Symmetry and periodicity
  • Addition theorems 3.2
  • 3.3 relationships
  • 3.4 derivation
  • 3.5 Differential Equation
  • 4.1 sinh
  • 4.2 cosh

Definition

Definition of the exponential function

Therefore, the hyperbolic functions are periodic ( with purely imaginary period). The power series of cosh ( z) and sinh ( z) arise from those of cos ( z ) and sin (z ) by any negative characters are replaced by plus signs.

The name hyperbolic functions derived from the fact that they can be used for parameterization of the hyperbola:

In analogy to the circuit which can be parameterized by the sine and cosine: x = cos (t) and y = sin (t).

The functions provide a connection between the area A, which is enclosed by a line starting from the neutral point and its mirror image on the x - axis and the hyperbola, and the length of different routes.

Here, sinh ( A), the (positive ) Y- coordinate of the intersection with the straight line of the hyperbola and cosh ( A) the corresponding x -coordinate. tanh ( A) is the y coordinate of the line at x = 1, that is the slope of the straight line.

If one calculates the area by integration, we obtain the representation by means of the exponential function.

Properties of the real hyperbolic

For all real numbers are even and real.

The real function is strictly increasing and has 0 in its single turning point.

The real function is strictly decreasing for values

Due to apply all the properties of the complex hyperbolic functions are listed in the following paragraph on the real - limited functions.

Properties of the complex hyperbolic

For all complex numbers shall:

Symmetry and periodicity

  • , That is, sinh is an odd function.
  • , That is, cosh is an even function.
  • ,

That is, it is purely " imaginary frequency " before with minimal period length.

Addition theorems

Relationships

Derivation

The derivation of the hyperbolic sine is:

The derivation of the hyperbolic cosine is:

The derivative of the hyperbolic tangent is:

Differential equation

The functions and form as a base and the ( linear ) differential equation

Prompts you in general for the two basic solutions of this differential equation 2nd order yet and so they are already defined by sinh and cosh clearly. Say, this property can also be used as a definition of these two hyperbolic functions.

Bijectivity of the complex hyperbolic

Sinh

Let define the following subsets of the complex numbers:

Then the complex function is the " strip " A bijective onto B.

Cosh

Then the complex function is the " strip " A bijective onto B.

Alternative Names

  • For the hyperbolic functions, the name hyperbolic functions is common.
  • For the names hyperbolic sine and sine are hsin, hyperbolic use.
  • For the names of HCO, Hyperbelcosinus and cosine are used hyperbolic. The graph corresponds to the chain line ( catenoid ).

Derived functions

  • Hyperbolic tangent
  • Hyperbolic cotangent
  • Hyperbolic secant
  • Hyperbolic cosecant

Conversion table

Inverse functions

The inverse hyperbolic functions of the hot area functions. See also: related to the circular functions

Domain (mathematical analysis) Holomorphic function Catenary
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