C++ Standard Library<complex>

In mathematics and engineering computations, we often need to deal with complex numbers, such as in signal processing, Fourier transforms, circuit analysis, etc.

The C++ standard library provides<complex>The header file allows you to operate on complex numbers as easily as ordinary numbers.

A complex number consists of a real part and an imaginary part, in the form:

z = a + bi

A complex number is a combination of a real number and an imaginary number, usually represented asa + bi, whereais the real part,bis the imaginary part,iis the imaginary unit, satisfyingi^2 = -1。

In C++, the complex type isstd::complex<T>represented, whereTcan be any arithmetic type, such asfloat、doubleorlong double。

To use<complex>For the library, you first need to include this header file in your C++ program:

#include <iostream>
#include <complex>

Example

#include <iostream>
#include <complex> // complex number header file

int main() {
    std::complex<double> z1(3.0, 4.0); // 3 + 4i
    std::complex<double> z2(1.0, -2.0); // 1 - 2i

    std::cout << "z1 = " << z1 << std::endl;
    std::cout << "z2 = " << z2 << std::endl;
}

Output:

z1 = (3,4)
z2 = (1,-2)

Basic Syntax

Create complex numbers

std::complex<double> c(5.0, 3.0); // 创建一个复数 5 + 3i

Accessing Real and Imaginary Parts

double realPart = c.real(); // 获取实部
double imagPart = c.imag(); // 获取虚部

Basic Operations on Complex Numbers

C++ Standard Library<complex>supports the following basic operations:

  • Addition:operator+
  • Subtraction:operator-
  • Multiplication:operator*
  • Division:operator/
  • Conjugate:conj
  • Modulus:abs
  • Argument:arg

1. Getting the real and imaginary parts

std::cout << "实部: " << z1.real() << std::endl; // 3
std::cout << "虚部: " << z1.imag() << std::endl; // 4

2. Four Basic Arithmetic Operations:

Example

auto z3 = z1 + z2;   // (4, 2)
auto z4 = z1 - z2;   // (2, 6)
auto z5 = z1 * z2;   // (11, -2)
auto z6 = z1 / z2;   // (-1, 2)

std::cout << "z1 + z2 = " << z3 << std::endl;
std::cout << "z1 * z2 = " << z5 << std::endl;

3. Common Functions

Header file<complex>provides many mathematical functions related to complex numbers.

Example

#include <cmath> // Some math functions needed

std::cout << "Modulus |z1| = " << std::abs(z1) << std::endl;     // 5
std::cout << "Argument arg(z1) = " << std::arg(z1) << std::endl;  // 0.927 (radians)
std::cout << "Conjugate conjugate(z1) = " << std::conj(z1) << std::endl; // (3,-4)
std::cout << "exp(z1) = " << std::exp(z1) << std::endl;       // e^(3+4i)
std::cout << "sin(z1) = " << std::sin(z1) << std::endl;
std::cout << "cos(z1) = " << std::cos(z1) << std::endl;

4. Polar Coordinate Representation

Sometimes we need to represent complex numbers in polar coordinates (magnitude + phase angle).

Example

// Create complex number from polar coordinates
double r = 5.0;           // Modulus
double theta = M_PI / 4;  // 45 degrees
std::complex<double> z = std::polar(r, theta);

std::cout << "Polar complex number z = " << z << std::endl;  // (3.53553,3.53553)

5. Template Parameters

std::complex<T>It is a template class that supports different numeric types:

  • std::complex<float>
  • std::complex<double>(commonly used)
  • std::complex<long double>

Example

std::complex<float>  zf(1.0f, 2.0f);
std::complex<double> zd(1.0, 2.0);

Example

Below is an example using<complex>A simple example of the header file, including creating complex numbers, basic operations, and output results.

Example

#include <iostream>
#include <complex>

int main() {
    // Create two complex numbers
    std::complex<double> c1(5.0, 3.0);   // 5 + 3i
    std::complex<double> c2(2.0, -4.0);  // 2 - 4i

    // Output complex numbers
    std::cout << "c1: " << c1 << std::endl;  // (5,3)
    std::cout << "c2: " << c2 << std::endl;  // (2,-4)

    // Complex number addition
    std::complex<double> sum = c1 + c2;
    std::cout << "Sum: " << sum << std::endl;  // 7 - i

    // Complex number subtraction
    std::complex<double> diff = c1 - c2;
    std::cout << "Difference: " << diff << std::endl;  // 3 + 7i

    // Complex number multiplication
    std::complex<double> product = c1 * c2;
    std::cout << "Product: " << product << std::endl;  // 22 - 14i

    // Complex number division
    std::complex<double> quotient = c1 / c2;
    std::cout << "Quotient: " << quotient << std::endl;  // -0.1 + 1.3i

    // Complex conjugate
    std::complex<double> conjugate = std::conj(c1);
    std::cout << "Conjugate of c1: " << conjugate << std::endl;  // 5 - 3i

    // Complex modulus
    double modulus = std::abs(c1);
    std::cout << "Modulus of c1: " << modulus << std::endl;  // sqrt(34) ≈ 5.83095

    // Complex argument (radians)
    double argument = std::arg(c1);
    std::cout << "Argument of c1: " << argument << std::endl;  // atan(3/5) ≈ 0.54042 rad

    return 0;
}

When you run the above program, you will get the following output:

c1: (5,3)
c2: (2,-4)
Sum: (7,-1)
Difference: (3,7)
Product: (22,-14)
Quotient: (-0.1,1.3)
Conjugate of c1: (5,-3)
Modulus of c1: 5.83095
Argument of c1: 0.54042
other extensions