org.netlib.math.complex
Class Complex

java.lang.Object
  extended by org.netlib.math.complex.Complex
All Implemented Interfaces:
java.io.Serializable, java.lang.Cloneable
Direct Known Subclasses:
SimpleNode

public class Complex
extends java.lang.Object
implements java.lang.Cloneable, java.io.Serializable

See Also:
Serialized Form

Field Summary
static java.lang.String AUTHOR
           
static java.lang.String DATE
           
static Complex iii
          A constant representing i, the famous square root of -1.
 double im
           
 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.
 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.
 java.lang.Object clone()
          Overrides the Cloneable interface.
 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) .
 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.
 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.
 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
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

iii

public static final Complex iii
A constant representing i, the famous square root of -1.

The other square root of -1 is - i.


re

public double re

im

public 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:

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


clone

public java.lang.Object clone()
Overrides the Cloneable interface.

Standard override; no change in semantics.

The following Java code example illustrates how to clone, or copy, a Complex number:

     Complex z1 =  new Complex(0, 1);
     Complex z2 =  (Complex) z1.clone();
 

Overrides:
clone in class java.lang.Object
Returns:
An Object that is a copy of this Complex object.

See Also:
Cloneable, Object.clone()

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()

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:

Domain Restrictions:

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


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:

Domain Restrictions:

See Also:
scale(double)

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:

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

See Also:
pow(Complex, double)

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:

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:

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:

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:

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:

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:

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:

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

     System.out.println("z = " + z);
 
would print something like
     z = (2.0 - 5.0i)
 

Overrides:
toString in class java.lang.Object
Returns:
String containing the cartesian coordinate representation

See Also:
cart(double, double)


Fraid (system and language). (C) 2003-2007 Ivaylo Iliev
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.