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# Complex

Works with complex numbers where each component is of some specified type
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### Key Facts

Gyroscopic Couple: The rate of change of angular momentum () = (In the limit).
• = Moment of Inertia.
• = Angular velocity
• = Angular velocity of precession.

Blaise Pascal (1623-1662) was a French mathematician, physicist, inventor, writer and Catholic philosopher.

Leonhard Euler (1707-1783) was a pioneering Swiss mathematician and physicist.

## Definition

The complex template class is defined in the standard header <complex>, and in the nonstandard backward-compatibility header <complex.h>.
namespace std {
template <class T>
class complex;
}
The template parameter T is used as the scalar type of both the real and the imaginary parts of the complex number.

## Description

The template class std::complex operates on complex numbers.

A complex number is a number which can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where

## Complex Numbers Operations

Create, Copy, and Assign Operations

Operation Effect
complex c Creates a complex number with 0 as the real part and 0 as the imaginary part
complex c(1.1) Creates a complex number with 1.1 as the real part and 0 as the imaginary part
complex c(1.1,2.2) Creates a complex number with 1.1 as the real part and 1.2 as the imaginary part
complex c1(c2) Creates c1 as a copy of c2
polar (2. 2) Creates a temporary complex number from polar coordinates (2.2 as magnitude rho and 0 as phase angle theta)
polar (2. 2, 0.77) Creates a temporary complex number from polar coordinates (2.2 as magnitude rho and 0.77 as phase angle theta)
conj (c) Creates a temporary complex number that is the conjugated complex number of c
c1 = c2 Assigns the values of c2 to c1
c1 += c2 Adds the value of c2 to c1
c1 -= c2 Subtracts the value of c2 from c1
c1 *= c2 Multiplies the value of c2 by c1
c1 /= c2 Divides the value of c2 into c1

Value Access

Operation Effect
c.real() Returns the value of the real part (member function)
real(c) Returns the value of the real part (global function)
c.imag() Returns the value of the imaginary part (member function)
imag(c) Returns the value of the imaginary part (global function)
abs(c) Returns the absolute value of c
norm(c) Returns the squared absolute value of c(c.real()^2 + c.imag()^2)
arg(c) Returns the angle of the polar representation of c

Comparison Operations

Operation Effect
c1 == c2 Returns if c1 is equal to c2
c1 != c2 Returns if c1 differs from c2

Arithmetic Operations

Operation Effect
c1 + c2 Returns the sum of c1 and c2
c1 - c2 Returns the difference between c1 and c2
c1 * c2 Returns the product of c1 and c2
c1 / c2 Returns the quotient of c1 and c2
-c Returns the negated value of c
+ c Returns c
c1 += c2 Same with c1 = c1 + c2
c1 -= c2 Same with c1 = c1 - c2
c1 *= c2 Same with c1 = c1 * c2
c1 /= c2 Same with c1 = c1 / c2

Input/Output Operations

Operation Effect
strm << c Writes the complex number c to the ostream strm
strm >> c Reads the complex number c from the istream strm

Transcendental Functions

Operation Effect
pow(c1, c2) Complex power c1^c2
exp(c) Base e exponential of c (e^c)
sqrt(c) Square root of c
log(c) Complex natural logarithm of c with base e (ln c)
log10(c) Complex common logarithm of c with base 10 (lg c)
sin(c) Sine of c
cos(c) Cosine of c
tan(c) Tangent of c
sinh(c) Hyperbolic sine of c
cosh(c) Hyperbolic cosine of c
tanh(c) Hyperbolic tangent of c

### References:

• Nicolai M. Josuttis: "The C++ Standard Library"

Example:
##### Example - Complex numbers
Problem
The following program performs some common operations on complex numbers.
Workings
#include <iostream>
#include <complex>
using namespace std;

int main()
{
/*complex number with real and imaginary parts
*-real part: 4.0
*-imaginary part: 3.0
*/
complex<double> c1(4.0,3.0);

/*create complex number from polar coordinates
*-magnitude:5.0
*-phase angle:0.75
*/
complex<float> c2(polar(5.0,0.75));

// print complex numbers with real and imaginary parts
cout << "c1: " << c1 << endl;
cout << "c2: " << c2 << endl;

//print complex numbers as polar coordinates
cout << "c1: magnitude: " << abs (c1)
<< " (squared magnitude: " << norm(c1) << ") "
<< " phase angle: " << arg(c1) << endl;
cout << "c2: magnitude: " << abs(c2)
<< " (squared magnitude: " << norm (c2) << ") "
<< " phase angle: " << arg(c2) << endl;

//print complex conjugates
cout << "c1 conjugated: " << conj(c1) << endl;
cout << "c2 conjugated: " << conj(c2) << endl;

//print result of a computation
cout << "4.4 + c1 * 1.8: " << 4.4 + c1 * 1.8 << endl;

/*print sum of c1 and c2:
*-note: different types
*/
cout << "c1 + c2:        "
<< c1 + complex<double>(c2.real(),c2.imag()) << endl;

// add square root of c1 to c1 and print the result
cout << "c1 += sqrt(c1): " << (c1 += sqrt(c1)) << endl;

return 0;
}
Solution
Output:

c1: (4,3)
c2: (3.65844,3.40819)
c1: magnitude: 5 (squared magnitude: 25) phase angle: 0.643501
c2: magnitude: 5 (squared magnitude: 25) phase angle: 0.75
c1 conjugated: (4,-3)
c2 conjugated: (3.65844,-3.40819)
4.4 + c1 * 1.8: (11.6,5.4)
c1 + c2: (7.65844,6.40819)
c1 += sqrt(c1): (6.12132,3.70711)
References
• Nicolai M. Josuttis: "The C++ Standard Library"