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#### cpp_complex

`#include <boost/multiprecision/cpp_complex.hpp>`

    namespace boost{ namespace multiprecision{

       template <unsigned Digits, backends::digit_base_type DigitBase = backends::digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
       using cpp_complex_backend = complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >;

       template <unsigned Digits, backends::digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0, expression_template_option ExpressionTemplates = et_off>
       using cpp_complex = number<complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >, ExpressionTemplates>;

       typedef cpp_complex<50> cpp_complex_50;
       typedef cpp_complex<100> cpp_complex_100;

       typedef cpp_complex<24, backends::digit_base_2, void, std::int16_t, -126, 127> cpp_complex_single;
       typedef cpp_complex<53, backends::digit_base_2, void, std::int16_t, -1022, 1023> cpp_complex_double;
       typedef cpp_complex<64, backends::digit_base_2, void, std::int16_t, -16382, 16383> cpp_complex_extended;
       typedef cpp_complex<113, backends::digit_base_2, void, std::int16_t, -16382, 16383> cpp_complex_quad;
       typedef cpp_complex<237, backends::digit_base_2, void, std::int32_t, -262142, 262143> cpp_complex_oct;

    }} // namespaces

The `cpp_complex_backend` back-end is used in conjunction with `number`: It acts as an entirely C++ (header only and dependency free) complex number type that is a drop-in replacement for `std::complex`, but with much greater precision.

The template alias `cpp_complex` avoids the need to use class `number` directly.

Type `cpp_complex` can be used at fixed precision by specifying a non-zero `Digits` template parameter. The typedefs `cpp_complex_50` and `cpp_complex_100` provide complex number types at 50 and 100 decimal digits precision respectively.

Optionally, you can specify whether the precision is specified in decimal digits or binary bits - for example to declare a `cpp_complex` with exactly the same precision as `std::complex<double>` one would use `cpp_complex<53, digit_base_2>`. The typedefs `cpp_complex_single`, `cpp_complex_double`, `cpp_complex_quad`, `cpp_complex_oct` and `cpp_complex_double_extended` provide software analogues of the IEEE single, double, quad and octuple float data types, plus the Intel-extended-double type respectively. Note that while these types are functionally equivalent to the native IEEE types, but they do not have the same size or bit-layout as true IEEE compatible types.

Normally `cpp_complex` allocates no memory: all of the space required for its digits are allocated directly within the class. As a result care should be taken not to use the class with too high a digit count as stack space requirements can grow out of control. If that represents a problem then providing an allocator as a template parameter causes `cpp_complex` to dynamically allocate the memory it needs: this significantly reduces the size of `cpp_complex` and increases the viable upper limit on the number of digits at the expense of performance. However, please bear in mind that arithmetic operations rapidly become very expensive as the digit count grows: the current implementation really isn't optimized or designed for large digit counts. Note that since the actual type of the objects allocated is completely opaque, the suggestion would be to use an allocator with `char` `value_type`, for example: `cpp_complex<1000, digit_base_10, std::allocator<char> >`.

The next template parameters determine the type and range of the exponent: parameter `Exponent` can be any signed integer type, but note that `MinExponent` and `MaxExponent` can not go right up to the limits of the `Exponent` type as there has to be a little extra headroom for internal calculations. You will get a compile time error if this is the case. In addition if MinExponent or MaxExponent are zero, then the library will choose suitable values that are as large as possible given the constraints of the type and need for extra headroom for internal calculations.

Finally, as with class `number`, the final template parameter determines whether expression templates are turn on or not. Since by default this type allocates no memory, expression template support is off by default. However, you should probably turn it on if you specify an allocator.

There is full standard library support available for this type, comparable with what `std::complex` provides.

Things you should know when using this type:

- Default constructed `cpp_complex`s have a value of zero.
- The radix of this type is 2, even when the precision is specified as decimal digits.
- The type supports both infinities and NaNs. An infinity is generated whenever the result would overflow, and a NaN is generated for any mathematically undefined operation.
- There is no `std::numeric_limits` specialisation for this type: this is the same behaviour as `std::complex`. If you need `std::numeric_limits` support you need to look at `std::numeric_limits<my_complex_number_type::value_type>`.
- Any `number` instantiated on this type, is convertible to any other `number` instantiated on this type - for example you can convert from `number<cpp_complex<50> >` to `number<cpp_bin_float<SomeOtherValue> >`. Narrowing conversions round to nearest and are `explicit`.
- Conversion from a string results in a `std::runtime_error` being thrown if the string can not be interpreted as a valid complex number.

###### example:

    #include <iostream>
    #include <complex>
    #include <boost/multiprecision/cpp_complex.hpp>

    template<class Complex>
    void complex_number_examples()
    {
        Complex z1{0, 1};
        std::cout << std::setprecision(std::numeric_limits<typename Complex::value_type>::digits10);
        std::cout << std::scientific << std::fixed;
        std::cout << "Print a complex number: " << z1 << std::endl;
        std::cout << "Square it             : " << z1*z1 << std::endl;
        std::cout << "Real part             : " << z1.real() << " = " << real(z1) << std::endl;
        std::cout << "Imaginary part        : " << z1.imag() << " = " << imag(z1) << std::endl;
        using std::abs;
        std::cout << "Absolute value        : " << abs(z1) << std::endl;
        std::cout << "Argument              : " << arg(z1) << std::endl;
        std::cout << "Norm                  : " << norm(z1) << std::endl;
        std::cout << "Complex conjugate     : " << conj(z1) << std::endl;
        std::cout << "Projection onto Riemann sphere: " <<  proj(z1) << std::endl;
        typename Complex::value_type r = 1;
        typename Complex::value_type theta = 0.8;
        using std::polar;
        std::cout << "Polar coordinates (phase = 0)    : " << polar(r) << std::endl;
        std::cout << "Polar coordinates (phase !=0)    : " << polar(r, theta) << std::endl;

        std::cout << "\nElementary special functions:\n";
        using std::exp;
        std::cout << "exp(z1) = " << exp(z1) << std::endl;
        using std::log;
        std::cout << "log(z1) = " << log(z1) << std::endl;
        using std::log10;
        std::cout << "log10(z1) = " << log10(z1) << std::endl;
        using std::pow;
        std::cout << "pow(z1, z1) = " << pow(z1, z1) << std::endl;
        using std::sqrt;
        std::cout << "Take its square root  : " << sqrt(z1) << std::endl;
        using std::sin;
        std::cout << "sin(z1) = " << sin(z1) << std::endl;
        using std::cos;
        std::cout << "cos(z1) = " << cos(z1) << std::endl;
        using std::tan;
        std::cout << "tan(z1) = " << tan(z1) << std::endl;
        using std::asin;
        std::cout << "asin(z1) = " << asin(z1) << std::endl;
        using std::acos;
        std::cout << "acos(z1) = " << acos(z1) << std::endl;
        using std::atan;
        std::cout << "atan(z1) = " << atan(z1) << std::endl;
        using std::sinh;
        std::cout << "sinh(z1) = " << sinh(z1) << std::endl;
        using std::cosh;
        std::cout << "cosh(z1) = " << cosh(z1) << std::endl;
        using std::tanh;
        std::cout << "tanh(z1) = " << tanh(z1) << std::endl;
        using std::asinh;
        std::cout << "asinh(z1) = " << asinh(z1) << std::endl;
        using std::acosh;
        std::cout << "acosh(z1) = " << acosh(z1) << std::endl;
        using std::atanh;
        std::cout << "atanh(z1) = " << atanh(z1) << std::endl;
    }

    int main()
    {
        std::cout << "First, some operations we usually perform with std::complex:\n";
        complex_number_examples<std::complex<double>>();
        std::cout << "\nNow the same operations performed using quad precision complex numbers:\n";
        complex_number_examples<boost::multiprecision::cpp_complex_quad>();

        return 0;
    }

Which produces the output (for the multiprecision type):

    Print a complex number: (0.000000000000000000000000000000000,1.000000000000000000000000000000000)
    Square it             : -1.000000000000000000000000000000000
    Real part             : 0.000000000000000000000000000000000 = 0.000000000000000000000000000000000
    Imaginary part        : 1.000000000000000000000000000000000 = 1.000000000000000000000000000000000
    Absolute value        : 1.000000000000000000000000000000000
    Argument              : 1.570796326794896619231321691639751
    Norm                  : 1.000000000000000000000000000000000
    Complex conjugate     : (0.000000000000000000000000000000000,-1.000000000000000000000000000000000)
    Projection onto Riemann sphere: (0.000000000000000000000000000000000,1.000000000000000000000000000000000)
    Polar coordinates (phase = 0)    : 1.000000000000000000000000000000000
    Polar coordinates (phase !=0)    : (0.696706709347165389063740022772448,0.717356090899522792567167815703377)

    Elementary special functions:
    exp(z1) = (0.540302305868139717400936607442977,0.841470984807896506652502321630299)
    log(z1) = (0.000000000000000000000000000000000,1.570796326794896619231321691639751)
    log10(z1) = (0.000000000000000000000000000000000,0.682188176920920673742891812715678)
    pow(z1, z1) = 0.207879576350761908546955619834979
    Take its square root  : (0.707106781186547524400844362104849,0.707106781186547524400844362104849)
    sin(z1) = (0.000000000000000000000000000000000,1.175201193643801456882381850595601)
    cos(z1) = 1.543080634815243778477905620757062
    tan(z1) = (0.000000000000000000000000000000000,0.761594155955764888119458282604794)
    asin(z1) = (0.000000000000000000000000000000000,0.881373587019543025232609324979793)
    acos(z1) = (1.570796326794896619231321691639751,-0.881373587019543025232609324979793)
    atan(z1) = (0.000000000000000000000000000000000,inf)
    sinh(z1) = (0.000000000000000000000000000000000,0.841470984807896506652502321630299)
    cosh(z1) = 0.540302305868139717400936607442977
    tanh(z1) = (0.000000000000000000000000000000000,1.557407724654902230506974807458360)
    asinh(z1) = (0.000000000000000000000000000000000,1.570796326794896619231321691639751)
    acosh(z1) = (0.881373587019543025232609324979792,1.570796326794896619231321691639751)
    atanh(z1) = (0.000000000000000000000000000000000,0.785398163397448309615660845819876)