Computes complex arc hyperbolic cosine of a complex valuez with branch cut at values less than 1 along the real axis.

[edit]Parameters

[edit]Return value

If no errors occur, the complex arc hyperbolic cosine ofz is returned, in the range of a half-strip of nonnegative values along the real axis and in the interval[−iπ; +iπ] along the imaginary axis.

[edit]Error handling and special values

Errors are reported consistent withmath_errhandling.

If the implementation supports IEEE floating-point arithmetic,

• std::acosh(std::conj(z))==std::conj(std::acosh(z)).
• Ifz is(±0,+0), the result is(+0,π/2).
• Ifz is(x,+∞) (for any finite x), the result is(+∞,π/2).
• Ifz is(x,NaN) (for any^[1] finite x), the result is(NaN,NaN) andFE_INVALID may be raised.
• Ifz is(-∞,y) (for any positive finite y), the result is(+∞,π).
• Ifz is(+∞,y) (for any positive finite y), the result is(+∞,+0).
• Ifz is(-∞,+∞), the result is(+∞,3π/4).
• Ifz is(±∞,NaN), the result is(+∞,NaN).
• Ifz is(NaN,y) (for any finite y), the result is(NaN,NaN) andFE_INVALID may be raised.
• Ifz is(NaN,+∞), the result is(+∞,NaN).
• Ifz is(NaN,NaN), the result is(NaN,NaN).

1. ↑perC11 DR471, this holds for non-zero x only. Ifz is(0,NaN), the result should be(NaN,π/2).

[edit]Notes

Although the C++ standard names this function "complex arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic cosine", and, less common, "complex area hyperbolic cosine".

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment(-∞,+1) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic cosine isacosh z = ln(z +√z+1√z-1).

[edit]Example

#include <complex>#include <iostream> int main(){std::cout<<std::fixed;std::complex<double> z1(0.5,0);std::cout<<"acosh"<< z1<<" = "<<std::acosh(z1)<<'\n'; std::complex<double> z2(0.5,-0.0);std::cout<<"acosh"<< z2<<" (the other side of the cut) = "<<std::acosh(z2)<<'\n'; // in upper half-plane, acosh = i acosstd::complex<double> z3(1,1), i(0,1);std::cout<<"acosh"<< z3<<" = "<<std::acosh(z3)<<'\n'<<"i*acos"<< z3<<" = "<< i*std::acos(z3)<<'\n';}

acosh(0.500000,0.000000) = (0.000000,-1.047198)acosh(0.500000,-0.000000) (the other side of the cut) = (0.000000,1.047198)acosh(1.000000,1.000000) = (1.061275,0.904557)i*acos(1.000000,1.000000) = (1.061275,0.904557)

[edit]See also

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