The Complex Class

This chapter describes:

How to use complex numbers with the class complex
Functions in the library:
- Constructor function
- Operator functions
- Complex functions
- Trigonometric and hyperbolic functions
- Mathematical functions

Class complex is implemented as a set of overloaded operators and functions that perform general complex mathematics. A complex number has two data members: a real and an imaginary part. Both are of type double.

This class implements complex numbers. Using overloaded functions and operators, you can treat complex numbers like you treat reals or integers. You can use the following:

Arithmetic operators (+, -, /, *), relational operators (==, !=), and logical operators (&&, ||)
Stream operators << and >>.
Mathematical functions, such as sqrt() and log10().

Using the Complex Class

To use complex numbers, add the library complex to your project and include file oldcomplex.h in your file, in a statement like this:

#include <oldcomplex.h>

You can declare a complex variable three ways: uninitialized, initialized with two numbers, or initialized with another complex number. For example:

complex z1;		// Uninitialized
complex z2(1.0,2.0);	// Initialized with two numbers
complex z3 = z2;	// Initialized with a complex number

To access components of a complex number, use member functions complex::real() and complex::imag(). For example:

complex z1, z2(5.0,6.0);
z1.real() = z2.imag(); 
z1.imag() = z2.real();

To print or read a complex number, use stream operators operator>> and operator<<. For example, this program:

#include <complex.h>
#include <iostream.h>

int main()
{   
   complex z1(1.1, 2.0), z2; 
   cout << "Enter z2: "; 
   cin >> z2; 
   cout << "z1 = " << z1 << '\n'; 
   cout << "z2 = " << z2 << '\n'; 
   return 0; 
}

prints this:

Enter z2: 10 9.8 
z1 = (1.1, 2) 
z2 = (10,9.8)

You can also use a complex number in the middle of a statement. For example, this line prints out the sine of (1, 1):

cout << sin(complex(1,1));

Note: To invoke the complex version of a function, one of its arguments must be complex. For example, to figure the square root of -1, you must use sqrt(complex(-1,0)).

Member Functions

The five types of class member functions are:

Constructors
Operators
Complex
Trigonometric and hyperbolic
Mathematical

Constructor Function

The following function is a constructor function.

complex::complex()

Header

oldcomplex.h

Prototype

complex::complex(); complex::complex(double re, double im); complex::complex(const complex& z);

Description

The first constructor creates an unitialized complex number. For example:

complex z;

The second constructor creates a complex number with the real part initialized to re and the imaginary part initialized to im. For example:

complex z1(1.2,3.0), z2(3,4);

The third constructor creates a copy of a complex number. It is also called the X(X&) constructor. For example:

#include <oldcomplex.h>

complex complexFunc() 
{ 
// Invokes complex(double, double)
   complex z1(2.0,4.0); 

// Invokes complex(const complex&) 
   complex z2 = z1; 

// Invokes complex(const complex&) to 
// create a temporary variable and adds 
// it to z1. 
   return complex(1.1, 2.2) + z1; 
}

Operator Functions

The operators defined for complex are overloaded to have the same meaning as the built-in operators. For example, + is the addition of two complex variables just as it is used for the addition of two variables of type double. The operators in the complex class have the usual operator precedence.

The five types of operators defined for the complex class are:

arithmetic	operator+ operator- operator* operator/
relational	operator&& operator\|\| operator! operator==
assignment	complex::operator= complex::operator+= complex::operator-= complex::operator* complex::operator/
unary	complex::operator- complex::operator!
stream	operator<< operator>>

complex::operator=

Header: oldcomplex.h
Prototype: complex& complex::operator=(const complex& z); complex& complex::operator=(double u);
Description: Assigns a value to complex variable.

complex::operator+=

Header: oldcomplex.h
Prototype: complex& complex::operator+=(const complex& z); complex& complex::operator+=(double u);
Description: Adds a value to a complex variable and assigns the result to the complex variable.

complex::operator-=

Header: oldcomplex.h
Prototype: complex& complex::operator-=(const complex& z); complex& complex::operator-=(double u);
Description: Subtracts a value from a complex variable and assigns the result to the complex variable.

complex::operator*=

Header: oldcomplex.h
Prototype: complex& complex::operator*=(const complex& z); complex& complex::operator*=(double u);
Description: Multiplies a value by a complex variable and assigns the result to the complex variable.

complex::operator/=

Header: oldcomplex.h
Prototype: complex& complex::operator/=(const complex& z); complex& complex::operator/=(double u);
Description: Divides a complex variable by a value and assigns the result to the complex variable.

complex::operator-

Header: oldcomplex.h
Prototype: complex complex::operator-() const;
Description: Returns the negative of a complex number.

complex::operator!

Header: oldcomplex.h
Prototype: int complex::operator!() const;
Description: Performs logical negation on a complex number.

operator+

Header: oldcomplex.h
Prototype: complex operator+(const complex& z1, complex& z2); complex operator+(double u, const complex& z); complex operator+(const complex& z, double u);
Description: Returns the sum of a complex number and any other number.

operator-

Header: oldcomplex.h
Prototype: complex operator-(const complex& z1, const complex& z2); complex operator-(double u, const complex& z); complex operator-(const complex& z, double u);
Description: Returns the difference of a complex number and any other number.

operator*

Header: oldcomplex.h
Prototype: complex operator*(const complex& z1, const complex& z2); complex operator*(double u, const complex& z); complex operator*(const complex& z, double u);
Description: Returns the product of a complex number and any other number.

operator/

Header: oldcomplex.h
Prototype: complex operator/(const complex& z1, const complex& z2); complex operator/(double u, const complex& z); complex operator/(const complex& z, double u);
Description: Returns the quotient of a complex number and any other number.

operator&&

Header: oldcomplex.h
Prototype: complex operator&&(const complex& z1, const complex& z2); complex operator&&(double u, const complex& z); complex operator&&(const complex& z, double u);
Description: Performs logical AND. Returns 0 if both arguments are 0 or (0, 0).

operator||

Header: oldcomplex.h
Prototype: complex operator||(const complex& z1, const complex& z2); complex operator||(double u, const complex& z); complex operator||(const complex& z, double u);
Description: Performs logical OR. Returns 0 if both arguments are 0 or (0, 0).

operator!

Header: oldcomplex.h
Prototype: complex operator!=(const complex& z1, const complex& z2); complex operator!=(double u, const complex& z); complex operator!=(const complex& z, double u);
Description: Tests for inequality.

operator==

Header: oldcomplex.h
Prototype: complex ::operator==(const complex& z1, const complex& z2); complex ::operator==(double u, const complex& z); complex ::operator==(const complex& z, double u);
Description: Tests for equality.

operator<<

Header: oldcomplex.h
Prototype: ostream& operator<<(ostream& s, const complex& z);
Description: Reads z from the stream s, in the form(real, imaginary).

operator>>

Header: oldcomplex.h
Prototype: istream& operator>>(istream& s, const complex& z);
Description: Reads z from the stream s, in the form(real, imaginary).

Complex Functions

The complex functions are unique to the complex type and do not overload names of the ANSI C math library.

arg

Header

oldcomplex.h

Prototype

double arg(const complex& z);

Description

The arg function determines the angle, in radians, of the number in
the complex plane specified in the x argument. The positive real axis
has angle 0; the positive imaginary axis has angle pi/2.

The arg function is equivalent to:

atan2(imag(x), real(x))

Return Value

Returns the argument of the complex number z in radians.

complex::imag, complex::real

Header

oldcomplex.h

Prototype

double& complex::imag(); double& complex::real();

Description

complex::imag returns the imaginary portion of a complex number; complex::real returns the real portion. Either function can be on the left side or right side of an assignment statement.

Example

For complex::imag 		For complex::real 

#include <oldcomplex.h> 	#include <oldcomplex.h> 

complex z1(1.0,2.0), z2; 	complex z1(1.0,2.0), z2;
z1.imag() = z2.imag(); 		z1.real() = z2.real();

conj

Header: oldcomplex.h
Prototype: complex conj(const complex& z);
Description: Returns the complex conjugate of a complex number. If the complex number z is x+iy, then conj(z) is x-iy.

imag

Header: oldcomplex.h
Prototype: double imag(const complex& z);
Description: Returns the imaginary portion of a complex number. It can be on only the right side of an assignment statement.

modulus

Header: oldcomplex.h
Prototype: double modulus(const complex& z);
Description: Returns the absolute value of a complex number. The absolute value of a complex number is also called the modulus. modulus(z) is equal to abs(z), which is sqrt (x * x + y * y).

norm

Header: oldcomplex.h
Prototype: double norm(const complex& z);
Description: Returns the square of the absolute value of a complex number. For z, of form (x + iy), conj(z) = (x * x + y * y).

polar

Header

oldcomplex.h

Prototype

complex polar(double range, double theta);

Description

Returns the complex number that is the result of converting polar coordinates range and theta to x and y coordinates. This is equivalent to using the formula:

complex(range * cos(theta), range * sin(theta));

real

Header: oldcomplex.h
Prototype: double real(const complex& z);
Description: Returns the real portion of a complex number. It can be on only the right side of an assignment statement.

Trigonometric and hyperbolic functions

The trigonometric and hyperbolic functions overload the names of ANSI C library floating-point trigonometric and hyperbolic functions.

acos

Header: oldcomplex.h
Prototype: complex acos(const complex& z);
Description: Returns the arccosine of a complex number.

asin, asinh

Header: oldcomplex.h
Prototype: complex asin(const complex& z); complex asinh(const complex& z);
Description: asin returns the arcsine of a complex number. asinh returns the hyperbolic arcsine of a complex number.

atan, atanh

Header: oldcomplex.h
Prototype: complex atan(const complex& z); complex atanh(const complex& z);
Description: atan returns the arctangent of a complex number. atanh returns the hyperbolic arctangent of a complex number.

cos, cosh

Header: oldcomplex.h
Prototype: complex cos(const complex& z); complex cosh(const complex& z);
Description: cos returns the cosine of a complex number. cosh returns the hyperbolic cosine of a complex number.

sin, sinh

Header: oldcomplex.h
Prototype: complex sin (const complex& z); complex sinh(const complex& z);
Description: sin returns the sine of a complex number. sinh returns the hyperbolic sine of a complex number.

tan, tanh

Header: oldcomplex.h
Prototype: complex tan(const complex& z); complex tanh(const complex& z);
Description: tan returns the tangent of a complex number. tanh returns the hyperbolic tangent of a complex number.

Mathematical functions

The mathematical functions overload functions contained in the ANSI C math library to provide consistent complex support.

abs

Header: oldcomplex.h
Prototype: double abs(const complex&);
Description: Returns the absolute value of a complex number.

exp

Header: oldcomplex.h
Prototype: complex exp(const complex&);
Description: Returns the value of e raised to a complex number.

log, log10

Header: oldcomplex.h
Prototype: complex log(const complex&); complex log10(const complex&);
Description: log returns the logarithm base e of a complex number. log10 returns the logarithm base 10 of a complex number.

pow

Header: oldcomplex.h
Prototype: complex pow(const complex&, double); complex pow(const complex&, const complex &); complex pow(double, const complex&); complex pow(const complex&, int);
Description: Returns any number raised to a complex number or vice versa.

sqrt

Header: oldcomplex.h
Prototype: complex sqrt(const complex&);
Description: Returns the square root of a complex number.

Standard C

Standard C++

Win32

DOS, DOS32, Win16

Other C

Other C++

Using the Complex Class

Member Functions

Constructor Function

Operator Functions

Complex Functions

Trigonometric and hyperbolic functions

Mathematical functions