Inverse hyperbolic functions

Definition of Inverse Hyperbolic Functions

• Inverse hyperbolic functions are the inverses of the hyperbolic functions sinh, cosh, and tanh, and are often denoted as arcsinh, arccosh, and arctanh respectively.
• Arcsinh(x) is the function defined by the equation y = arcsinh(x) if and only if x = sinh(y).
• Similarly, arccosh(x) and arctanh(x) are respectively defined by the equations y = arccosh(x) if and only if x = cosh(y), and y = arctanh(x) if and only if x = tanh(y).
• Like hyperbolic function, inverse hyperbolic functions are also functions of real variables.

Properties of Inverse Hyperbolic Functions

• Unlike their hyperbolic counterparts, inverse hyperbolic functions do not possess symmetry properties because they are neither odd nor even.
• Inverse hyperbolic functions can be expressed in terms of logarithms:
  • arcsinh(x) = ln(x + sqrt(1 + x^2))
  • arccosh(x) = ln(x + sqrt(x^2 - 1)) {only defined for x≥1}

  • arctanh(x) = 0.5 * ln((1 + x) / (1 - x)) {only defined for

    x

    <1}

• The function arcsinh(x) is an increasing function, while arccosh(x) is a decreasing function.
• The composition of a function and its inverse is the identity function, for example cosh(arcsinh(x)) = sqrt(1 + x^2) and sinh(arccosh(x)) = sqrt(x^2 - 1)

Calculating with Inverse Hyperbolic Functions

• Inverse hyperbolic functions, like regular hyperbolic functions, can be manipulated using standard rules of algebra and calculus.
• The derivatives of the inverse hyperbolic functions can be derived as follows:
  • (d/dx) arcsinh(x) = 1 / sqrt(1 + x^2)
  • (d/dx) arccosh(x) = 1 / sqrt(x^2 - 1) {only defined for x>1}

  • (d/dx) arctanh(x) = 1 / (1 - x^2) {only defined for

    x

    <1}

• The integrals of the inverse hyperbolic functions can be derived using integration by parts or other common methods.