std::asinh(std::complex)
Defined in header <complex>
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template< class T > complex<T> asinh( const complex<T>& z ); |
(since C++11) | |
Computes complex arc hyperbolic sine of a complex value z with branch cuts outside the interval [−i; +i] along the imaginary axis.
Contents |
[edit] Parameters
z | - | complex value |
[edit] Return value
If no errors occur, the complex arc hyperbolic sine of z 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 with math_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)
- If z is
(+0,+0)
, the result is(+0,+0)
- If z is
(x,+∞)
(for any positive finite x), the result is(+∞,π/2)
- If z is
(x,NaN)
(for any 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(+∞,+0)
- If z is
(+∞,+∞)
, the result is(+∞,π/4)
- If z is
(+∞,NaN)
, the result is(+∞,NaN)
- If z is
(NaN,+0)
, the result is(NaN,+0)
- If z is
(NaN,y)
(for any finite nonzero y), the result is(NaN,NaN)
and FE_INVALID may be raised - If z is
(NaN,+∞)
, the result is(±∞,NaN)
(the sign of the real part is unspecified) - If z 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 is asinh z = ln(z + √1+z2
).
asin(iz) |
i |
[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) / i std::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'; }
Output:
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
(C++11) |
computes area hyperbolic cosine of a complex number (arcosh(z)) (function template) |
(C++11) |
computes area hyperbolic tangent of a complex number (artanh(z)) (function template) |
computes hyperbolic sine of a complex number (sinh(z)) (function template) | |
(C++11)(C++11)(C++11) |
computes the inverse hyperbolic sine (arsinh(x)) (function) |
C documentation for casinh
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