# Standard library header <complex>

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This header is part of the numeric library.

### Classes

complex | a complex number type (class template)

### Functions

##### Operations

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)
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)

##### Literals

operator""ifoperator""ioperator""il (C++14) | a std::complex literal representing purely imaginary number (function)

### Synopsis

    namespace std {
        template<class T> class complex;

        template<> class complex<float>;
        template<> class complex<double>;
        template<> class complex<long double>;

        // operators:
        template<class T> constexpr complex<T> operator+(
            const complex<T>&, const complex<T>&);
        template<class T> constexpr complex<T> operator+(const complex<T>&, const T&);
        template<class T> constexpr complex<T> operator+(const T&, const complex<T>&);

        template<class T> constexpr complex<T> operator-(
            const complex<T>&, const complex<T>&);
        template<class T> constexpr complex<T> operator-(const complex<T>&, const T&);
        template<class T> constexpr complex<T> operator-(const T&, const complex<T>&);

        template<class T> constexpr complex<T> operator*(
            const complex<T>&, const complex<T>&);
        template<class T> constexpr complex<T> operator*(const complex<T>&, const T&);
        template<class T> constexpr complex<T> operator*(const T&, const complex<T>&);

        template<class T> constexpr complex<T> operator/(
            const complex<T>&, const complex<T>&);
        template<class T> constexpr complex<T> operator/(const complex<T>&, const T&);
        template<class T> constexpr complex<T> operator/(const T&, const complex<T>&);

        template<class T> constexpr complex<T> operator+(const complex<T>&);
        template<class T> constexpr complex<T> operator-(const complex<T>&);

        template<class T> constexpr bool operator==(const complex<T>&, const complex<T>&);
        template<class T> constexpr bool operator==(const complex<T>&, const T&);
        template<class T> constexpr bool operator==(const T&, const complex<T>&);

        template<class T> constexpr bool operator!=(const complex<T>&, const complex<T>&);
        template<class T> constexpr bool operator!=(const complex<T>&, const T&);
        template<class T> constexpr bool operator!=(const T&, const complex<T>&);

        template<class T, class CharT, class Traits>
        basic_istream<CharT, Traits>&
        operator>>(basic_istream<CharT, Traits>&, complex<T>&);

        template<class T, class CharT, class Traits>
        basic_ostream<CharT, Traits>&
        operator<<(basic_ostream<CharT, Traits>&, const complex<T>&);

        // values:
        template<class T> constexpr T real(const complex<T>&);
        template<class T> constexpr T imag(const complex<T>&);

        template<class T> T abs(const complex<T>&);
        template<class T> T arg(const complex<T>&);
        template<class T> constexpr T norm(const complex<T>&);

        template<class T> constexpr complex<T> conj(const complex<T>&);
        template<class T> complex<T> proj(const complex<T>&);
        template<class T> complex<T> polar(const T&, const T& = 0);

        // transcendentals:
        template<class T> complex<T> acos(const complex<T>&);
        template<class T> complex<T> asin(const complex<T>&);
        template<class T> complex<T> atan(const complex<T>&);

        template<class T> complex<T> acosh(const complex<T>&);
        template<class T> complex<T> asinh(const complex<T>&);
        template<class T> complex<T> atanh(const complex<T>&);

        template<class T> complex<T> cos  (const complex<T>&);
        template<class T> complex<T> cosh (const complex<T>&);
        template<class T> complex<T> exp  (const complex<T>&);
        template<class T> complex<T> log  (const complex<T>&);
        template<class T> complex<T> log10(const complex<T>&);

        template<class T> complex<T> pow(const complex<T>&, const T&);
        template<class T> complex<T> pow(const complex<T>&, const complex<T>&);
        template<class T> complex<T> pow(const T&, const complex<T>&);

        template<class T> complex<T> sin (const complex<T>&);
        template<class T> complex<T> sinh(const complex<T>&);
        template<class T> complex<T> sqrt(const complex<T>&);
        template<class T> complex<T> tan (const complex<T>&);
        template<class T> complex<T> tanh(const complex<T>&);

        // tuple interface:
        template<class T> struct tuple_size;
        template<size_t I, class T> struct tuple_element;
        template<class T> struct tuple_size<complex<T>>;
        template<size_t I, class T> struct tuple_element<I, complex<T>>;
        template<size_t I, class T>
          constexpr T& get(complex<T>&) noexcept;
        template<size_t I, class T>
          constexpr T&& get(complex<T>&&) noexcept;
        template<size_t I, class T>
          constexpr const T& get(const complex<T>&) noexcept;
        template<size_t I, class T>
          constexpr const T&& get(const complex<T>&&) noexcept;

        // complex literals:
        inline namespace literals {
            inline namespace complex_literals {
                constexpr complex<long double> operator""il(long double);
                constexpr complex<long double> operator""il(unsigned long long);
                constexpr complex<double> operator""i(long double);
                constexpr complex<double> operator""i(unsigned long long);
                constexpr complex<float> operator""if(long double);
                constexpr complex<float> operator""if(unsigned long long);
            }
        }
    }

#### Class std::complex

    template<class T>
    class complex {
    public:
        typedef T value_type;
        constexpr complex(const T& re = T(), const T& im = T());
        constexpr complex(const complex&) = default;
        template<class X> constexpr explicit(/* see constructor page */)
            complex(const complex<X>&);

        constexpr T real() const;
        constexpr void real(T);
        constexpr T imag() const;
        constexpr void imag(T);

        constexpr complex<T>& operator= (const T&);
        constexpr complex<T>& operator+=(const T&);
        constexpr complex<T>& operator-=(const T&);
        constexpr complex<T>& operator*=(const T&);
        constexpr complex<T>& operator/=(const T&);

        constexpr complex& operator=(const complex&);
        template<class X> constexpr complex<T>& operator= (const complex<X>&);
        template<class X> constexpr complex<T>& operator+=(const complex<X>&);
        template<class X> constexpr complex<T>& operator-=(const complex<X>&);
        template<class X> constexpr complex<T>& operator*=(const complex<X>&);
        template<class X> constexpr complex<T>& operator/=(const complex<X>&);
    };

### 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 79 | C++98 | the default argument of the second parameter of `polar` was missing in the synopsis | added

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