std::acosh(std::complex)
| Defined in header <complex>
|
||
| template< class T > complex<T> acosh( const complex<T>& z ); |
(since C++11) | |
Computes complex arc hyperbolic sine of a complex value z with branch cut at values less than 1 along the real axis.
Contents |
[edit] Parameters
| z | - | complex value |
[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
zis(±0,+0), the result is(+0,π/2) - If
zis(x,+∞)(for any finite x), the result is(+∞,π/2) - If
zis(x,NaN)(for any[1] finite x), the result is(NaN,NaN)and FE_INVALID may be raised. - If
zis(-∞,y)(for any positive finite y), the result is(+∞,π) - If
zis(+∞,y)(for any positive finite y), the result is(+∞,+0) - If
zis(-∞,+∞), the result is(+∞,3π/4) - If
zis(±∞,NaN), the result is(+∞,NaN) - If
zis(NaN,y)(for any finite y), the result is(NaN,NaN)and FE_INVALID may be raised. - If
zis(NaN,+∞), the result is(+∞,NaN) - If
zis(NaN,NaN), the result is(NaN,NaN)
[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 sine is acosh z = ln(z + √z+1 √z-1)
For any z, acosh(z) =| √z-1 |
| √1-z |
[edit] Example
#include <iostream> #include <complex> 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 acos std::complex<double> z3(1, 1), i(0, 1); std::cout << "acosh" << z3 << " = " << std::acosh(z3) << '\n' << "i*acos" << z3 << " = " << i*std::acos(z3) << '\n'; }
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
| (C++11) |
computes arc cosine of a complex number (arccos(z)) (function template) |
| (C++11) |
computes area hyperbolic sine of a complex number (function template) |
| (C++11) |
computes area hyperbolic tangent of a complex number (function template) |
| computes hyperbolic cosine of a complex number (ch(z)) (function template) | |
| (C++11) |
computes the inverse hyperbolic cosine (arcosh(x)) (function) |
| C documentation for cacosh
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