std::asinh(std::complex)(3)

💅 std::asinh(std::complex) 💅

okay so welcome to the C++ Standard Libary manual, section 3, bestie. This is the page for std::asinh(std::complex)(3) — the complex-number arc-hyperbolic-sine situation — and we're gonna read it together like a group chat. no cap. ✨

💖 NAME 💖

std::asinh(std::complex) - std::asinh(std::complex). That's the name, literally the same thing twice; it's giving "yes, the complex version, keep up." 🫶

✨ Synopsis ✨

Defined in header <complex>
template< class T >                       (since C++11)
complex<T> asinh( const complex<T>& z );

It computes the complex arc hyperbolic sine of a complex value z, with branch cuts outside the interval [−i; +i] along the imaginary axis. 🌀

🧚 Parameters 🧚

• z — complex value. That's the whole parameter list, bestie. iconic minimalism. 💅

🌸 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. 🦋

🚨 Error handling and special values 🚨

Errors are reported consistent with math_errhandling.

If the implementation supports IEEE floating-point arithmetic, then:

• 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)

🦋 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
).

For any z, asinh(z) = asin(iz) / i. 🧮

🌈 Example 🌈

       // Run this code

        #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)

💅 See also 💅

• acosh(std::complex) — computes area hyperbolic cosine of a complex number (C++11) (\({\small\operatorname{arcosh}{z}}\)arcosh(z)) (function template)
• atanh(std::complex) — computes area hyperbolic tangent of a complex number (C++11) (\({\small\operatorname{artanh}{z}}\)artanh(z)) (function template)
• sinh(std::complex) — computes hyperbolic sine of a complex number (C++11) (\({\small\sinh{z}}\)sinh(z)) (function template)
• asinh — computes the inverse hyperbolic sine (C++11) (\({\small\operatorname{arsinh}{x}}\)arsinh(x)) (function)
• asinhf — computes the inverse hyperbolic sine (C++11) (\({\small\operatorname{arsinh}{x}}\)arsinh(x)) (function)
• asinhl — computes the inverse hyperbolic sine (C++11) (\({\small\operatorname{arsinh}{x}}\)arsinh(x)) (function)
• C documentation for casinh

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okay that's the whole page, bestie. sourced from http://cppreference.com, version/date 2024.06.10, and yes it's the std::asinh(std::complex)(3) page from the C++ Standard Libary manual. mwah 💋✨