std::acosh(std::complex) - cppreference.com (original) (raw)

| | | | | --------------------------------------------------------------- | | ------------- | | template< class T > complex<T> acosh( const complex<T>& z ); | | (since C++11) |

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

Contents

• 1 Parameters
• 2 Return value
• 3 Error handling and special values
• 4 Notes
• 5 Example
• 6 See also

[edit] Parameters

[edit] Return value

If no errors occur, the complex arc hyperbolic cosine of z 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 with math_errhandling.

If the implementation supports IEEE floating-point arithmetic,

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

1. ↑ per C11 DR471, this holds for non-zero x only. If z 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 is acosh z = ln(z + √z+1 √z-1).

For any z, acosh(z) = acos(z), or simply i acos(z) in the upper half of the complex plane.

[edit] Example

Output:

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