Computes complex arc hyperbolic sine of a complex valuez with branch cuts outside the interval[−i; +i] along the imaginary axis.

[edit]Parameters

[edit]Return value

If no errors occur, the complex arc hyperbolic sine ofz is returned, in the range of a strip mathematically unbounded along the real axis and in the interval[−iπ/2; +iπ/2] 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::asinh(std::conj(z))==std::conj(std::asinh(z))
• std::asinh(-z)==-std::asinh(z)
• Ifz is(+0,+0), the result is(+0,+0)
• Ifz is(x,+∞) (for any positive finite x), the result is(+∞,π/2)
• Ifz is(x,NaN) (for any finite x), the result is(NaN,NaN) andFE_INVALID may be raised
• Ifz is(+∞,y) (for any positive finite y), the result is(+∞,+0)
• Ifz is(+∞,+∞), the result is(+∞,π/4)
• Ifz is(+∞,NaN), the result is(+∞,NaN)
• Ifz is(NaN,+0), the result is(NaN,+0)
• Ifz is(NaN,y) (for any finite nonzero y), the result is(NaN,NaN) andFE_INVALID may be raised
• Ifz is(NaN,+∞), the result is(±∞,NaN) (the sign of the real part is unspecified)
• Ifz is(NaN,NaN), the result is(NaN,NaN)

[edit]Notes

Although the C++ standard names this function "complex arc hyperbolic sine", 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 sine", and, less common, "complex area hyperbolic sine".

Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments(-i∞,-i) and(i,i∞) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine isasinh z = ln(z +√1+z2
).

[edit]Example

#include <complex>#include <iostream> int main(){std::cout<<std::fixed;std::complex<double> z1(0.0,-2.0);std::cout<<"asinh"<< z1<<" = "<<std::asinh(z1)<<'\n'; std::complex<double> z2(-0.0,-2);std::cout<<"asinh"<< z2<<" (the other side of the cut) = "<<std::asinh(z2)<<'\n'; // for any z, asinh(z) = asin(iz) / istd::complex<double> z3(1.0,2.0);std::complex<double> i(0.0,1.0);std::cout<<"asinh"<< z3<<" = "<<std::asinh(z3)<<'\n'<<"asin"<< z3* i<<" / i = "<<std::asin(z3* i)/ i<<'\n';}

asinh(0.000000,-2.000000) = (1.316958,-1.570796)asinh(-0.000000,-2.000000) (the other side of the cut) = (-1.316958,-1.570796)asinh(1.000000,2.000000) = (1.469352,1.063440)asin(-2.000000,1.000000) / i = (1.469352,1.063440)

[edit]See also

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