Complex Inverse Hyperbolic Functions

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1) The notations

2) Definitions    

3) Derivatives of inverse hyperbolic functions

4) Indefinite integrals

1) The notations

For inverse hyperbolic functions, the notations sinh^-1 and cosh^-1 are often used for arcsinh and arccosh, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion.

The inverse hyperbolic functions are the multivalued function that are the inverse functions of the hyperbolic functions.

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2) Definitions

The inverse hyperbolic functions may be expressed using natural logarithms.

arcsinh(z) = ln(z + )

arccosh(z) = ln() For real x > 1, this simplifies to arccosh(x) = ln(x + )

arctanh(z) = [ln(1 + z) - ln(1 - z)] For real x < 1, this simplifies to arctanh(x) = ln()

arccoth(z) = [ln(1 + ) - ln(1 - )] For real x < 0 or x >1, this simplifies to arccoth(x) = ln()

arccsch(z) = ln() For real x, it satisfies arccsch(x)=

arcsech(z) = ln() For real x, it satisfies arcsech(x)=

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3) Derivatives of inverse hyperbolic functions

arcsinh(z) =

arccosh(z) =

arctanh(z) =

arccoth(z) =

arccsch(z) = -

arcsech(z) = -

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4) Indefinite integrals

sinh-1zdz = zsinh-1z - + C

cosh-1zdz = zcosh-1z - (z+1)+ C

tanh-1zdz = ztanh-1z + ln(z2 - 1) + C

coth-1zdz = zcoth-1z + ln(z2 - 1) + C

csch-1zdz = zcsch-1z + ln(z (1 + )) + C

sech-1zdz = zsech-1z - tan-1() + C

 

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by Tetyana Butler