# std::complex

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< cpp‎ | numeric

Defined in header `<complex>` |  |
template< class T > class complex; | (1) |
template<> class complex<float>; | (2) | (until C++23)
template<> class complex<double>; | (3) | (until C++23)
template<> class complex<long double>; | (4) | (until C++23)

Specializations of `std::complex` for cv-unqualified standard(until C++23) floating-point types are TriviallyCopyable (since C++23) LiteralTypes for representing and manipulating complex number.

### Template parameters

T | - | the type of the real and imaginary parts. The behavior is unspecified (and may fail to compile) if `T` is not a cv-unqualified standard(until C++23) floating-point type and undefined if `T` is not NumericType.

### Member types

Member type | Definition
`value_type` | `T`

### Member functions

(constructor) | constructs a complex number (public member function)
operator= | assigns the contents (public member function)
real | accesses the real part of the complex number (public member function)
imag | accesses the imaginary part of the complex number (public member function)
operator+=operator-=operator*=operator/= | compound assignment of two complex numbers or a complex and a scalar (public member function)

### Non-member functions

operator+operator- | applies unary operators to complex numbers (function template)
operator+operator-operator*operator/ | performs complex number arithmetic on two complex values or a complex and a scalar (function template)
operator==operator!= (removed in C++20) | compares two complex numbers or a complex and a scalar (function template)
operator<<operator>> | serializes and deserializes a complex number (function template)
get(std::complex) (C++26) | obtains a reference to real or imaginary part from a std::complex (function template)
real | returns the real part (function template)
imag | returns the imaginary part (function template)
abs(std::complex) | returns the magnitude of a complex number (function template)
arg | returns the phase angle (function template)
norm | returns the squared magnitude (function template)
conj | returns the complex conjugate (function template)
proj (C++11) | returns the projection onto the Riemann sphere (function template)
polar | constructs a complex number from magnitude and phase angle (function template)

##### Exponential functions

exp(std::complex) | complex base e exponential (function template)
log(std::complex) | complex natural logarithm with the branch cuts along the negative real axis (function template)
log10(std::complex) | complex common logarithm with the branch cuts along the negative real axis (function template)

##### Power functions

pow(std::complex) | complex power, one or both arguments may be a complex number (function template)
sqrt(std::complex) | complex square root in the range of the right half-plane (function template)

##### Trigonometric functions

sin(std::complex) | computes sine of a complex number (\({\small\sin{z}}\)sin(z)) (function template)
cos(std::complex) | computes cosine of a complex number (\({\small\cos{z}}\)cos(z)) (function template)
tan(std::complex) | computes tangent of a complex number (\({\small\tan{z}}\)tan(z)) (function template)
asin(std::complex) (C++11) | computes arc sine of a complex number (\({\small\arcsin{z}}\)arcsin(z)) (function template)
acos(std::complex) (C++11) | computes arc cosine of a complex number (\({\small\arccos{z}}\)arccos(z)) (function template)
atan(std::complex) (C++11) | computes arc tangent of a complex number (\({\small\arctan{z}}\)arctan(z)) (function template)

##### Hyperbolic functions

sinh(std::complex) | computes hyperbolic sine of a complex number (\({\small\sinh{z}}\)sinh(z)) (function template)
cosh(std::complex) | computes hyperbolic cosine of a complex number (\({\small\cosh{z}}\)cosh(z)) (function template)
tanh(std::complex) | computes hyperbolic tangent of a complex number (\({\small\tanh{z}}\)tanh(z)) (function template)
asinh(std::complex) (C++11) | computes area hyperbolic sine of a complex number (\({\small\operatorname{arsinh}{z}}\)arsinh(z)) (function template)
acosh(std::complex) (C++11) | computes area hyperbolic cosine of a complex number (\({\small\operatorname{arcosh}{z}}\)arcosh(z)) (function template)
atanh(std::complex) (C++11) | computes area hyperbolic tangent of a complex number (\({\small\operatorname{artanh}{z}}\)artanh(z)) (function template)

### Helper types

std::tuple_size<std::complex> (C++26) | obtains the size of a std::complex (class template specialization)
std::tuple_element<std::complex> (C++26) | obtains the underlying real and imaginary number type of a std::complex (class template specialization)

### Array-oriented access

For any object z of type `std::complex<T>`, reinterpret_cast<T(&)[2]>(z)[0] is the real part of z and reinterpret_cast<T(&)[2]>(z)[1] is the imaginary part of z.

For any pointer to an element of an array of `std::complex<T>` named p and any valid array index i, reinterpret_cast<T*>(p)[2 * i] is the real part of the complex number p[i], and reinterpret_cast<T*>(p)[2 * i + 1] is the imaginary part of the complex number p[i].

The intent of this requirement is to preserve binary compatibility between the C++ library complex number types and the C language complex number types (and arrays thereof), which have an identical object representation requirement.

### Implementation notes

In order to satisfy the requirements of array-oriented access, an implementation is constrained to store the real and imaginary parts of a `std::complex` specialization in separate and adjacent memory locations. Possible declarations for its non-static data members include:

- an array of type `value_type[2]`, with the first element holding the real part and the second element holding the imaginary part (e.g. Microsoft Visual Studio);
- a single member of type `value_type _Complex` (encapsulating the corresponding C language complex number type) (e.g. GNU libstdc++);
- two members of type `value_type`, with the same member access, holding the real and the imaginary parts respectively (e.g. LLVM libc++).

An implementation cannot declare additional non-static data members that would occupy storage disjoint from the real and imaginary parts, and must ensure that the class template specialization does not contain any padding bit. The implementation must also ensure that optimizations to array access account for the possibility that a pointer to `value_type` may be aliasing a `std::complex` specialization or array thereof.

### Literals

Defined in inline namespace `std::literals::complex_literals`
operator""ifoperator""ioperator""il (C++14) | a std::complex literal representing purely imaginary number (function)

### Notes

Feature-test macro | Value | Std | Feature
--- | --- | --- | ---
`__cpp_lib_constexpr_complex` | `201711L` | (C++20) | constexpr simple complex mathematical functions in `<complex>`
`202306L` | (C++26) | More constexpr for `<complex>`
`__cpp_lib_tuple_like` | `202311L` | (C++26) | Add tuple protocol to `std::complex`

### Example

Run this code

    #include <cmath>
    #include <complex>
    #include <iomanip>
    #include <iostream>
    #include <ranges>

    int main()
    {
        using namespace std::complex_literals;
        std::cout << std::fixed << std::setprecision(1);

        std::complex<double> z1 = 1i * 1i; // imaginary unit squared
        std::cout << "i * i = " << z1 << '\n';

        std::complex<double> z2 = std::pow(1i, 2); // imaginary unit squared
        std::cout << "pow(i, 2) = " << z2 << '\n';

        const double PI = std::acos(-1); // or std::numbers::pi in C++20
        std::complex<double> z3 = std::exp(1i * PI); // Euler's formula
        std::cout << "exp(i * pi) = " << z3 << '\n';

        std::complex<double> z4 = 1.0 + 2i, z5 = 1.0 - 2i; // conjugates
        std::cout << "(1 + 2i) * (1 - 2i) = " << z4 * z5 << '\n';

        const auto zz = {0.0 + 1i, 2.0 + 3i, 4.0 + 5i};
    #if __cpp_lib_tuple_like >= 202311L
        for (double re : zz | std::views::keys)
            std::cout << re << ' ';
        std::cout << '\n';
        for (double im : zz | std::views::values)
            std::cout << im << ' ';
        std::cout << '\n';
    #else
        for (double re : zz | std::views::transform([](auto z){ return z.real(); }))
            std::cout << re << ' ';
        std::cout << '\n';
        for (double im : zz | std::views::transform([](auto z){ return z.imag(); }))
            std::cout << im << ' ';
        std::cout << '\n';
    #endif
    }

Output:

    i * i = (-1.0,0.0)
    pow(i, 2) = (-1.0,0.0)
    exp(i * pi) = (-1.0,0.0)
    (1 + 2i) * (1 - 2i) = (5.0,0.0)
    0.0 2.0 4.0
    1.0 3.0 5.0

### Defect reports

The following behavior-changing defect reports were applied retroactively to previously published C++ standards.

DR | Applied to | Behavior as published | Correct behavior
--- | --- | --- | ---
LWG 387 | C++98 | `std::complex` was not guaranteed to be compatible with C `complex` | guaranteed to be compatible

### See also

C documentation for Complex number arithmetic

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