Class Complex

java.lang.Object
  Complex

All Implemented Interfaces:

java.lang.Cloneable, java.io.Serializable

public class Complex

extends java.lang.Object

implements java.lang.Cloneable, java.io.Serializable

Version:

1.0 FINAL
ALM Fri 29-Aug-97

A Java class for performing complex number arithmetic to double precision.

See Also:

Serialized Form

Field Summary

static java.lang.String	AUTHOR
static java.lang.String	DATE
static Complex	i A constant representing i, the famous square root of -1.
private double	im
private double	re
static java.lang.String	REMARK
protected static double	TWO_PI Twice PI radians is the same thing as 360 degrees.
static java.lang.String	VERSION

Constructor Summary

Complex()
Constructs a Complex representing the number zero.

Complex(Complex z)
Constructs a separate new Complex from an existing Complex.

Complex(double re)
Constructs a Complex representing a real number.

Complex(double re, double im)
Constructs a Complex from real and imaginary parts.

Method Summary

double	abs() Returns the magnitude of a Complex number.
private static double	abs(double x, double y)
Complex	acos() Returns the principal arc cosine of a Complex number.
Complex	acosh() Returns the principal inverse hyperbolic cosine of a Complex number.
Complex	add(Complex z) To perform z1 + z2, you write z1.add(z2) .
double	arg() Returns the principal angle of a Complex number, in radians, measured counter-clockwise from the real axis.
Complex	asin() Returns the principal arc sine of a Complex number.
Complex	asinh() Returns the principal inverse hyperbolic sine of a Complex number.
Complex	atan() Returns the principal arc tangent of a Complex number.
Complex	atanh() Returns the principal inverse hyperbolic tangent of a Complex number.
static Complex	cart(double re, double im) Returns a Complex from real and imaginary parts.
Complex	conj() Returns the Complex "conjugate".
Complex	cos() Returns the cosine of a Complex number.
Complex	cosec() Returns the cosecant of a Complex number.
Complex	cosh() Returns the hyperbolic cosine of a Complex number.
Complex	cot() Returns the cotangent of a Complex number.
Complex	div(Complex z) To perform z1 / z2, you write z1.div(z2) .
private static void	div(Complex z, double x, double y)
boolean	equals(Complex z, double tolerance) Decides if two Complex numbers are "sufficiently" alike to be considered equal.
Complex	exp() Returns the number e "raised to" a Complex power.
double	im() Extracts the imaginary part of a Complex as a double.
private static void	inv(Complex z) Returns the Complex multiplicative inverse.
boolean	isInfinite() Returns true if either the real or imaginary component of this Complex is an infinite value.
boolean	isNaN() Returns true if either the real or imaginary component of this Complex is a Not-a-Number (NaN) value.
Complex	log() Returns the principal natural logarithm of a Complex number.
static void	main(java.lang.String[] args) Useful for checking up on the exact version.
Complex	mul(Complex z) To perform z1 * z2, you write z1.mul(z2) .
Complex	neg() Returns the "negative" of a Complex number.
double	norm() Returns the square of the "length" of a Complex number.
static Complex	polar(double r, double theta) Returns a Complex from a size and direction.
Complex	pow(Complex exponent) Returns this Complex raised to the power of a Complex exponent.
static Complex	pow(Complex base, Complex exponent) Returns the Complex base raised to the power of the Complex exponent.
static Complex	pow(Complex base, double exponent) Returns the Complex base raised to the power of the exponent.
static Complex	pow(double base, Complex exponent) Returns the base raised to the power of the Complex exponent.
double	re() Extracts the real part of a Complex as a double.
static Complex	real(double real) Returns a Complex representing a real number.
Complex	scale(double scalar) Returns the Complex scaled by a real number.
Complex	sec() Returns the secant of a Complex number.
Complex	sin() Returns the sine of a Complex number.
Complex	sinh() Returns the hyperbolic sine of a Complex number.
Complex	sqrt() Returns a Complex representing one of the two square roots.
private static void	sqrt(Complex z)
Complex	sub(Complex z) To perform z1 - z2, you write z1.sub(z2) .
Complex	tan() Returns the tangent of a Complex number.
Complex	tanh() Returns the hyperbolic tangent of a Complex number.
java.lang.String	toString() Converts a Complex into a String of the form (a + bi).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

VERSION

public static final java.lang.String VERSION

See Also:

Constant Field Values

DATE

public static final java.lang.String DATE

See Also:

Constant Field Values

AUTHOR

public static final java.lang.String AUTHOR

See Also:

Constant Field Values

REMARK

public static final java.lang.String REMARK

See Also:

Constant Field Values

TWO_PI

protected static final double TWO_PI

Twice PI radians is the same thing as 360 degrees.

See Also:

Constant Field Values

i

public static final Complex i

A constant representing i, the famous square root of -1.

The other square root of -1 is - i.

re

private double re

im

private double im

Constructor Detail

Complex

public Complex()

Constructs a Complex representing the number zero.

Complex

public Complex(double re)

Constructs a Complex representing a real number.

Parameters:

re - The real number

See Also:

real(double)

Complex

public Complex(Complex z)

Constructs a separate new Complex from an existing Complex.

Parameters:

z - A Complex number

Complex

public Complex(double re,
               double im)

Constructs a Complex from real and imaginary parts.

Note:

All methods in class Complex which deliver a Complex are written such that no intermediate Complex objects get generated. This means that you can easily anticipate the likely effects on garbage collection caused by your own coding.

Parameters:

re - Real part

im - Imaginary part

See Also:

cart(double, double), polar(double, double)

Method Detail

main

public static void main(java.lang.String[] args)

Useful for checking up on the exact version.

real

public static Complex real(double real)

Returns a Complex representing a real number.

Parameters:

real - The real number

Returns:

Complex representation of the real

See Also:

re(), cart(double, double)

cart

public static Complex cart(double re,
                           double im)

Returns a Complex from real and imaginary parts.

Parameters:

re - Real part

im - Imaginary part

Returns:

Complex from Cartesian coordinates

See Also:

re(), im(), polar(double, double), toString()

polar

public static Complex polar(double r,
                            double theta)

Returns a Complex from a size and direction.

Parameters:

r - Size

theta - Direction (in radians)

Returns:

Complex from Polar coordinates

See Also:

abs(), arg(), cart(double, double)

pow

public static Complex pow(Complex base,
                          double exponent)

Returns the Complex base raised to the power of the exponent.

Parameters:

base - The base "to raise"

exponent - The exponent "by which to raise"

Returns:

base "raised to the power of" exponent

See Also:

pow(double, Complex)

pow

public static Complex pow(double base,
                          Complex exponent)

Returns the base raised to the power of the Complex exponent.

Parameters:

base - The base "to raise"

exponent - The exponent "by which to raise"

Returns:

base "raised to the power of" exponent

See Also:

pow(Complex, Complex), exp()

pow

public static Complex pow(Complex base,
                          Complex exponent)

Returns the Complex base raised to the power of the Complex exponent.

Parameters:

base - The base "to raise"

exponent - The exponent "by which to raise"

Returns:

base "raised to the power of" exponent

See Also:

pow(Complex, double), pow(Complex)

isInfinite

public boolean isInfinite()

Returns true if either the real or imaginary component of this Complex is an infinite value.

Returns:

true if either component of the Complex object is infinite; false, otherwise.

isNaN

public boolean isNaN()

Returns true if either the real or imaginary component of this Complex is a Not-a-Number (NaN) value.

Returns:

true if either component of the Complex object is NaN; false, otherwise.

equals

public boolean equals(Complex z,
                      double tolerance)

Decides if two Complex numbers are "sufficiently" alike to be considered equal.

tolerance is the maximum magnitude of the difference between them before they are considered not equal.

Checking for equality between two real numbers on computer hardware is a tricky business. Try

    System.out.println((1.0/3.0 * 3.0));

and you'll see the nature of the problem! It's just as tricky with Complex numbers.

Realize that because of these complications, it's possible to find that the magnitude of one Complex number a is less than another, b, and yet a.equals(b, myTolerance) returns true. Be aware!

Parameters:

z - The Complex to compare with

tolerance - The tolerance for equality

Returns:

true, or false

re

public double re()

Extracts the real part of a Complex as a double.

     re(x + i*y)  =  x
 

Returns:

The real part

See Also:

im(), cart(double, double), real(double)

im

public double im()

Extracts the imaginary part of a Complex as a double.

     im(x + i*y)  =  y
 

Returns:

The imaginary part

See Also:

re(), cart(double, double)

norm

public double norm()

Returns the square of the "length" of a Complex number.

     norm(x + i*y)  =  x*x + y*y
 

Always non-negative.

Returns:

The norm

See Also:

abs()

abs

public double abs()

Returns the magnitude of a Complex number.

     abs(z)  =  sqrt(norm(z))
 

In other words, it's Pythagorean distance from the origin (0 + 0i, or zero).

The magnitude is also referred to as the "modulus" or "length".

Always non-negative.

Returns:

The magnitude (or "length")

See Also:

arg(), polar(double, double), norm()

abs

private static double abs(double x,
                          double y)

arg

public double arg()

Returns the principal angle of a Complex number, in radians, measured counter-clockwise from the real axis. (Think of the reals as the x-axis, and the imaginaries as the y-axis.)

There are infinitely many solutions, besides the principal solution. If A is the principal solution of arg(z), the others are of the form:

     A + 2*k*PI
 

where k is any integer.

arg() always returns a double between -PI and +PI.

Note:

2*PI radians is the same as 360 degrees.

Domain Restrictions:

There are no restrictions: the class defines arg(0) to be 0

Returns:

Principal angle (in radians)

See Also:

abs(), polar(double, double)

neg

public Complex neg()

Returns the "negative" of a Complex number.

     neg(a + i*b)  =  -a - i*b
 

The magnitude of the negative is the same, but the angle is flipped through PI (or 180 degrees).

Returns:

Negative of the Complex

See Also:

scale(double)

conj

public Complex conj()

Returns the Complex "conjugate".

     conj(x + i*y)  =  x - i*y
 

The conjugate appears "flipped" across the real axis.

Returns:

The Complex conjugate

inv

private static void inv(Complex z)

Returns the Complex multiplicative inverse.

     inv(z)  =  1 / z
 

Returns:

The Complex reciprocal

scale

public Complex scale(double scalar)

Returns the Complex scaled by a real number.

     scale((x + i*y), s)  =  (x*s + i*y*s)
 

Scaling by the real number 2.0, doubles the magnitude, but leaves the arg() unchanged. Scaling by -1.0 keeps the magnitude the same, but flips the arg() by PI (180 degrees).

Parameters:

scalar - A real number scale factor

Returns:

Complex scaled by a real number

See Also:

mul(Complex), div(Complex), neg()

add

public Complex add(Complex z)

To perform z1 + z2, you write z1.add(z2) .

     (a + i*b) + (c + i*d)  =  ((a+c) + i*(b+d))
 

sub

public Complex sub(Complex z)

To perform z1 - z2, you write z1.sub(z2) .

     (a + i*b) - (c + i*d)  =  ((a-c) + i*(b-d))
 

mul

public Complex mul(Complex z)

To perform z1 * z2, you write z1.mul(z2) .

     (a + i*b) * (c + i*d)  =  ( (a*c) - (b*d) + i*((a*d) + (b*c)) )
 

See Also:

scale(double)

div

public Complex div(Complex z)

To perform z1 / z2, you write z1.div(z2) .

     (a + i*b) / (c + i*d)  =  ( (a*c) + (b*d) + i*((b*c) - (a*d)) ) / norm(c + i*d)
 

Take care not to divide by zero!

Note:

Complex arithmetic in Java never causes exceptions. You have to deliberately check for overflow, division by zero, and so on, for yourself.

Domain Restrictions:

z1/z2 is undefined if z2 = 0

See Also:

scale(double)

div

private static void div(Complex z,
                        double x,
                        double y)

sqrt

public Complex sqrt()

Returns a Complex representing one of the two square roots.

     sqrt(z)  =  sqrt(abs(z)) * ( cos(arg(z)/2) + i * sin(arg(z)/2) )
 

For any complex number z, sqrt(z) will return the complex root whose arg is arg(z)/2.

Note:

There are always two square roots for each Complex number, except for 0 + 0i, or zero. The other root is the neg() of the first one. Just as the two roots of 4 are 2 and -2, the two roots of -1 are i and - i.

Returns:

The square root whose arg is arg(z)/2.

See Also:

pow(Complex, double)

sqrt

private static void sqrt(Complex z)

pow

public Complex pow(Complex exponent)

Returns this Complex raised to the power of a Complex exponent.

Parameters:

exponent - The exponent "by which to raise"

Returns:

this Complex "raised to the power of" the exponent

See Also:

pow(Complex, Complex)

exp

public Complex exp()

Returns the number e "raised to" a Complex power.

     exp(x + i*y)  =  exp(x) * ( cos(y) + i * sin(y) )
 

Note:

The value of e, a transcendental number, is roughly 2.71828182846...

Also, the following is quietly amazing:

     ePI*i    =    - 1
 

Returns:

e "raised to the power of" this Complex

See Also:

log(), pow(double, Complex)

log

public Complex log()

Returns the principal natural logarithm of a Complex number.

     log(z)  =  log(abs(z)) + i * arg(z)
 

There are infinitely many solutions, besides the principal solution. If L is the principal solution of log(z), the others are of the form:

     L + (2*k*PI)*i
 

where k is any integer.

Returns:

Principal Complex natural logarithm

See Also:

exp()

sin

public Complex sin()

Returns the sine of a Complex number.

     sin(z)  =  ( exp(i*z) - exp(-i*z) ) / (2*i)
 

Returns:

The Complex sine

See Also:

asin(), sinh(), cosec(), cos(), tan()

cos

public Complex cos()

Returns the cosine of a Complex number.

     cos(z)  =  ( exp(i*z) + exp(-i*z) ) / 2
 

Returns:

The Complex cosine

See Also:

acos(), cosh(), sec(), sin(), tan()

tan

public Complex tan()

Returns the tangent of a Complex number.

     tan(z)  =  sin(z) / cos(z)
 

Domain Restrictions:

tan(z) is undefined whenever z = (k + 1/2) * PI
where k is any integer

Returns:

The Complex tangent

See Also:

atan(), tanh(), cot(), sin(), cos()

cosec

public Complex cosec()

Returns the cosecant of a Complex number.

     cosec(z)  =  1 / sin(z)
 

Domain Restrictions:

cosec(z) is undefined whenever z = k * PI
where k is any integer

Returns:

The Complex cosecant

See Also:

sin(), sec(), cot()

sec

public Complex sec()

Returns the secant of a Complex number.

     sec(z)  =  1 / cos(z)
 

Domain Restrictions:

sec(z) is undefined whenever z = (k + 1/2) * PI
where k is any integer

Returns:

The Complex secant

See Also:

cos(), cosec(), cot()

cot

public Complex cot()

Returns the cotangent of a Complex number.

     cot(z)  =  1 / tan(z)
 

Domain Restrictions:

cot(z) is undefined whenever z = k * PI
where k is any integer

Returns:

The Complex cotangent

See Also:

tan(), cosec(), sec()

sinh

public Complex sinh()

Returns the hyperbolic sine of a Complex number.

     sinh(z)  =  ( exp(z) - exp(-z) ) / 2
 

Returns:

The Complex hyperbolic sine

See Also:

sin(), asinh()

cosh

public Complex cosh()

Returns the hyperbolic cosine of a Complex number.

     cosh(z)  =  ( exp(z) + exp(-z) ) / 2
 

Returns:

The Complex hyperbolic cosine

See Also:

cos(), acosh()

tanh

public Complex tanh()

Returns the hyperbolic tangent of a Complex number.

     tanh(z)  =  sinh(z) / cosh(z)
 

Returns:

The Complex hyperbolic tangent

See Also:

tan(), atanh()

asin

public Complex asin()

Returns the principal arc sine of a Complex number.

     asin(z)  =  -i * log(i*z + sqrt(1 - z*z))
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of asin(z), the others are of the form:

     k*PI + (-1)k  * A
 

where k is any integer.

Returns:

Principal Complex arc sine

See Also:

sin(), sinh()

acos

public Complex acos()

Returns the principal arc cosine of a Complex number.

     acos(z)  =  -i * log( z + i * sqrt(1 - z*z) )
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of acos(z), the others are of the form:

     2*k*PI +/- A
 

where k is any integer.

Returns:

Principal Complex arc cosine

See Also:

cos(), cosh()

atan

public Complex atan()

Returns the principal arc tangent of a Complex number.

     atan(z)  =  -i/2 * log( (i-z)/(i+z) )
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of atan(z), the others are of the form:

     A + k*PI
 

where k is any integer.

Domain Restrictions:

atan(z) is undefined for z = + i or z = - i

Returns:

Principal Complex arc tangent

See Also:

tan(), tanh()

asinh

public Complex asinh()

Returns the principal inverse hyperbolic sine of a Complex number.

     asinh(z)  =  log(z + sqrt(z*z + 1))
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of asinh(z), the others are of the form:

     k*PI*i + (-1)k  * A
 

where k is any integer.

Returns:

Principal Complex inverse hyperbolic sine

See Also:

sinh()

acosh

public Complex acosh()

Returns the principal inverse hyperbolic cosine of a Complex number.

     acosh(z)  =  log(z + sqrt(z*z - 1))
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of acosh(z), the others are of the form:

     2*k*PI*i +/- A
 

where k is any integer.

Returns:

Principal Complex inverse hyperbolic cosine

See Also:

cosh()

atanh

public Complex atanh()

Returns the principal inverse hyperbolic tangent of a Complex number.

     atanh(z)  =  1/2 * log( (1+z)/(1-z) )
 

There are infinitely many solutions, besides the principal solution. If A is the principal solution of atanh(z), the others are of the form:

     A + k*PI*i
 

where k is any integer.

Domain Restrictions:

atanh(z) is undefined for z = + 1 or z = - 1

Returns:

Principal Complex inverse hyperbolic tangent

See Also:

tanh()

toString

public java.lang.String toString()

Converts a Complex into a String of the form (a + bi).

This enables the Complex to be easily printed. For example, if z was 2 - 5i, then

     System.out.println("z = " + z);
 

would print

     z = (2 - 5i)
 

Returns:

String containing the cartesian coordinate representation

See Also:

cart(double, double)