Complex (JMSL Numerical Library)

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JMSL^TM Numerical Library 6.1

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com.imsl.math
Class Complex

java.lang.Object
  java.lang.Number
      com.imsl.math.Complex

All Implemented Interfaces:: Serializable, Cloneable

public class Complex
extends Number
implements Serializable, Cloneable
extends Number
implements Serializable, Cloneable

Set of mathematical functions for complex numbers. It provides the basic operations (addition, subtraction, multiplication, division) as well as a set of complex functions. The binary operations have the form, where op is add, subtract, multiply or divide.

        public static Complex op(Complex x, Complex y)   // x op y
        public static Complex op(Complex x, double y)    // x op y
        public static Complex op(double x, Complex y)    // x op y

Complex objects are immutable. Once created there is no way to change their value. The functions in this class follow the rules for complex arithmetic as defined C9x Annex G: IEC 559-compatible complex arithmetic. The API is not the same, but handling of infinities, NaNs, and positive and negative zeros is intended to follow the same rules.

See Also:: Example, Serialized Form

Field Summary
`static Complex`	`i` The imaginary unit.
`static String`	`suffix` String used in converting `Complex` to `String`.

Constructor Summary
`Complex()` Constructs a `Complex` equal to zero.
`Complex(Complex z)` Constructs a `Complex` equal to the argument.
`Complex(double re)` Constructs a `Complex` with a zero imaginary part.
`Complex(double re, double im)` Constructs a `Complex` with real and imaginary parts given by the input arguments.

Method Summary
`static double`	`abs(Complex z)` Returns the absolute value (modulus) of a `Complex`, \|z\|.
`static Complex`	`acos(Complex z)` Returns the inverse cosine (arc cosine) of a `Complex`, with branch cuts outside the interval [-1,1] along the real axis.
`static Complex`	`acosh(Complex z)` Returns the inverse hyperbolic cosine (arc cosh) of a `Complex`, with a branch cut at values less than one along the real axis.
`static Complex`	`add(Complex x, Complex y)` Returns the sum of two `Complex` objects, x+y.
`static Complex`	`add(Complex x, double y)` Returns the sum of a `Complex` and a `double`, x+y.
`static Complex`	`add(double x, Complex y)` Returns the sum of a `double` and a `Complex`, x+y.
`static double`	`argument(Complex z)` Returns the argument (phase) of a `Complex`, in radians, with a branch cut along the negative real axis.
`static Complex`	`asin(Complex z)` Returns the inverse sine (arc sine) of a `Complex`, with branch cuts outside the interval [-1,1] along the real axis.
`static Complex`	`asinh(Complex z)` Returns the inverse hyperbolic sine (arc sinh) of a `Complex`, with branch cuts outside the interval [-i,i].
`static Complex`	`atan(Complex z)` Returns the inverse tangent (arc tangent) of a `Complex`, with branch cuts outside the interval [-i,i] along the imaginary axis.
`static Complex`	`atanh(Complex z)` Returns the inverse hyperbolic tangent (arc tanh) of a `Complex`, with branch cuts outside the interval [-1,1] on the real axis.
`byte`	`byteValue()` Returns the value of the real part as a byte.
`int`	`compareTo(Complex z)` Compares two `Complex` objects.
`int`	`compareTo(Object obj)` Compares this `Complex` to another Object.
`static Complex`	`conjugate(Complex z)` Returns the complex conjugate of a `Complex` object.
`static Complex`	`cos(Complex z)` Returns the cosine of a `Complex`.
`static Complex`	`cosh(Complex z)` Returns the hyperbolic cosh of a `Complex`.
`static Complex`	`divide(Complex x, Complex y)` Returns the result of a `Complex` object divided by a `Complex` object, x/y.
`static Complex`	`divide(Complex x, double y)` Returns the result of a `Complex` object divided by a `double`, x/y.
`static Complex`	`divide(double x, Complex y)` Returns the result of a `double` divided by a `Complex` object, x/y.
`double`	`doubleValue()` Returns the value of the real part as a `double`.
`boolean`	`equals(Complex z)` Compares with another `Complex`.
`boolean`	`equals(Object obj)` Compares this object against the specified object.
`static Complex`	`exp(Complex z)` Returns the exponential of a `Complex` z, exp(z).
`float`	`floatValue()` Returns the value of the real part as a float.
`int`	`hashCode()` Returns a hashcode for this `Complex`.
`double`	`imag()` Returns the imaginary part of a `Complex` object.
`static double`	`imag(Complex z)` Returns the imaginary part of a `Complex` object.
`int`	`intValue()` Returns the value of the real part as an int.
`static Complex`	`log(Complex z)` Returns the logarithm of a `Complex` z, with a branch cut along the negative real axis.
`long`	`longValue()` Returns the value of the real part as a long.
`static Complex`	`multiply(Complex x, Complex y)` Returns the product of two `Complex` objects, x y*.
`static Complex`	`multiply(Complex x, double y)` Returns the product of a `Complex` object and a `double`, x y*.
`static Complex`	`multiply(double x, Complex y)` Returns the product of a `double` and a `Complex` object, x y*.
`static Complex`	`multiplyImag(Complex x, double y)` Returns the product of a `Complex` object and a pure imaginary `double`, x * iy.
`static Complex`	`multiplyImag(double x, Complex y)` Returns the product of a pure imaginary `double` and a `Complex` object, ix y*.
`static Complex`	`negate(Complex z)` Returns the negative of a `Complex` object, -z.
`static Complex`	`pow(Complex x, Complex y)` Returns the `Complex` x raised to the `Complex` y power.
`static Complex`	`pow(Complex z, double x)` Returns the `Complex` z raised to the x power, with a branch cut for the first parameter (z) along the negative real axis.
`double`	`real()` Returns the real part of a `Complex` object.
`static double`	`real(Complex z)` Returns the real part of a `Complex` object.
`short`	`shortValue()` Returns the value of the real part as a short.
`static Complex`	`sin(Complex z)` Returns the sine of a `Complex`.
`static Complex`	`sinh(Complex z)` Returns the hyperbolic sine of a `Complex`.
`static Complex`	`sqrt(Complex z)` Returns the square root of a `Complex`, with a branch cut along the negative real axis.
`static Complex`	`subtract(Complex x, Complex y)` Returns the difference of two `Complex` objects, x-y.
`static Complex`	`subtract(Complex x, double y)` Returns the difference of a `Complex` object and a `double`, x-y.
`static Complex`	`subtract(double x, Complex y)` Returns the difference of a `double` and a `Complex` object, x-y.
`static Complex`	`tan(Complex z)` Returns the tangent of a `Complex`.
`static Complex`	`tanh(Complex z)` Returns the hyperbolic tanh of a `Complex`.
`String`	`toString()` Returns a `String` representation for the specified `Complex`.
`static Complex`	`valueOf(String s)` Parses a `String` into a `Complex`.

Methods inherited from class java.lang.Object
`clone, finalize, getClass, notify, notifyAll, wait, wait, wait`

Field Detail

i

public static final Complex i

The imaginary unit. This constant is set to new Complex(0,1).

suffix

public static String suffix

String used in converting Complex to String. Default is i, but sometimes j is desired. Note that this is set for the class, not for a particular instance of a Complex.

Constructor Detail

Complex

public Complex()

Constructs a Complex equal to zero.

Complex

public Complex(Complex z)

Constructs a Complex equal to the argument.

Parameters:: z - a Complex object
Throws:: NullPointerException - is thrown if z is null

Complex

public Complex(double re)

Constructs a Complex with a zero imaginary part.

Parameters:: re - a double value equal to the real part of the Complex object

Complex

public Complex(double re,
               double im)

Constructs a Complex with real and imaginary parts given by the input arguments.

Parameters:: re - a double value equal to the real part of the Complex object; im - a double value equal to the imaginary part of the Complex object

Method Detail

abs

public static double abs(Complex z)

Returns the absolute value (modulus) of a Complex, |z|.

Parameters:: z - a Complex object
Returns:: a double value equal to the absolute value of the argument

acos

public static Complex acos(Complex z)

Returns the inverse cosine (arc cosine) of a Complex, with branch cuts outside the interval [-1,1] along the real axis.

Specifically, if z = x+iy,
.
returns .
returns .
returns .
returns , for finite x.
returns , for positive-signed finite y.
returns , for positive-signed finite y.
returns (where the sign of the imaginary part of the result is unspecified).
returns .
returns .
returns , for nonzero finite x.
returns , for finite y.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) cosine of the argument. The real part of the result is in the interval .

acosh

public static Complex acosh(Complex z)

Returns the inverse hyperbolic cosine (arc cosh) of a Complex, with a branch cut at values less than one along the real axis.

Specifically, if z = x+iy,
.
returns .
returns .
returns .
returns , for finite x.
returns , for positive-signed finite y.
returns , for positive-signed finite y.
returns .
returns .
returns , for finite x.
returns , for finite y.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) hyperbolic cosine of the argument. The real part of the result is non-negative and its imaginary part is in the interval .

add

public static Complex add(Complex x,
                          Complex y)

Returns the sum of two Complex objects, x+y.

Parameters:: x - a Complex object; y - a Complex object
Returns:: a newly constructed Complex initialized to x+y

add

public static Complex add(Complex x,
                          double y)

Returns the sum of a Complex and a double, x+y.

Parameters:: x - a Complex object; y - a double value
Returns:: a newly constructed Complex initialized to x+y

add

public static Complex add(double x,
                          Complex y)

Returns the sum of a double and a Complex, x+y.

Parameters:: x - a double value; y - a Complex object
Returns:: a newly constructed Complex initialized to x+y

argument

public static double argument(Complex z)

Returns the argument (phase) of a Complex, in radians, with a branch cut along the negative real axis.

Parameters:: z - a Complex object
Returns:: A double value equal to the argument (or phase) of a Complex. It is in the interval .

asin

public static Complex asin(Complex z)

Returns the inverse sine (arc sine) of a Complex, with branch cuts outside the interval [-1,1] along the real axis. The value of asin is defined in terms of the function asinh, by

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) sine of the argument. The real part of the result is in the interval .
See Also:: asinh(com.imsl.math.Complex)

asinh

public static Complex asinh(Complex z)

Returns the inverse hyperbolic sine (arc sinh) of a Complex, with branch cuts outside the interval [-i,i].

Specifically, if z = x+iy,
and asinh is odd.
returns .
returns .
returns for positive-signed finite x.
returns for positive-signed finite y.
returns (where the sign of the real part of the result is unspecified).
returns .
returns .
returns , for finite nonzero y.
returns , for finite x.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) hyperbolic sine of the argument. Its imaginary part is in the interval .

atan

public static Complex atan(Complex z)

Returns the inverse tangent (arc tangent) of a Complex, with branch cuts outside the interval [-i,i] along the imaginary axis. The value of atan is defined in terms of the function atanh, by

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) tangent of the argument. Its real part is in the interval .
See Also:: atanh(com.imsl.math.Complex)

atanh

public static Complex atanh(Complex z)

Returns the inverse hyperbolic tangent (arc tanh) of a Complex, with branch cuts outside the interval [-1,1] on the real axis.

Specifically, if z = x+iy,
and atanh is odd.
returns .
returns .
returns , for finite positive-signed y.
returns , for finite positive-signed x.
returns .
returns (where the sign of the real part of the result is unspecified).
returns .
returns , for finite y.
returns , for nonzero finite x.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the inverse (arc) hyperbolic tangent of the argument. The imaginary part of the result is in the interval .

byteValue

public byte byteValue()

Returns the value of the real part as a byte.

Overrides:: byteValue in class Number

Returns:: a byte representing the value of the real part of a Complex object

compareTo

public int compareTo(Complex z)

Compares two Complex objects.

A lexagraphical ordering is used. First the real parts are compared in the sense of Double.compareTo. If the real parts are unequal this is the return value. If the return parts are equal then the comparison of the imaginary parts is returned.

Parameters:: z - a Complex to be compared
Returns:: The value 0 if z is equal to this Complex; a value less than 0 if this Complex is less than z; and a value greater than 0 if this Complex is greater than z.

compareTo

public int compareTo(Object obj)

Compares this Complex to another Object. If the Object is a Complex, this function behaves like compareTo(Complex). Otherwise, it throws a ClassCastException (as Complex objects are comparable only to other Complex objects).

Parameters:: obj - an Object to be compared
Returns:: an int, 0 if obj is equal to this Complex; a value less than 0 if this Complex is less than obj; and a value greater than 0 if this Complex is greater than obj.
Throws:: ClassCastException - is thrown if obj is not a Complex object

conjugate

public static Complex conjugate(Complex z)

Returns the complex conjugate of a Complex object.

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the complex conjugate of Complex argument, z

cos

public static Complex cos(Complex z)

Returns the cosine of a Complex. The value of cos is defined in terms of the function cosh, by

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the cosine of the argument
See Also:: cosh(com.imsl.math.Complex)

cosh

public static Complex cosh(Complex z)

Returns the hyperbolic cosh of a Complex.

If z = x+iy,
and cosh is even.
returns .
returns (where the sign of the imaginary part of the result is unspecified).
returns .
returns .
returns , for finite nonzero x.
returns , for finite nonzero y.
returns (where the sign of the imaginary part of the result is unspecified).
returns .
returns , for finite nonzero x.
returns (where the sign of the imaginary part of the result is unspecified).
returns , for all nonzero numbers y.
returns .

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the hyperbolic cosine of the argument

divide

public static Complex divide(Complex x,
                             Complex y)

Returns the result of a Complex object divided by a Complex object, x/y.

Parameters:: x - a Complex object representing the numerator; y - a Complex object representing the denominator
Returns:: a newly constructed Complex initialized to x/y

divide

public static Complex divide(Complex x,
                             double y)

Returns the result of a Complex object divided by a double, x/y.

Parameters:: x - a Complex object representing the numerator; y - a double representing the denominator
Returns:: a newly constructed Complex initialized to x/y

divide

public static Complex divide(double x,
                             Complex y)

Returns the result of a double divided by a Complex object, x/y.

Parameters:: x - a double value; y - a Complex object representing the denominator
Returns:: a newly constructed Complex initialized to x/y

doubleValue

public double doubleValue()

Returns the value of the real part as a double.

Specified by:: doubleValue in class Number

Returns:: a double representing the value of the real part of a Complex object

equals

public boolean equals(Complex z)

Compares with another Complex.

Note: To be useful in hashtables this method considers two NaN double values to be equal. This is not according to IEEE specification.

Parameters:: z - a Complex object
Returns:: true if the real and imaginary parts of this object are equal to their counterparts in the argument; false, otherwise

equals

public boolean equals(Object obj)

Compares this object against the specified object.

Note: To be useful in hashtables this method considers two NaN double values to be equal. This is not according to IEEE specification

Overrides:: equals in class Object

Parameters:: obj - the object to compare with
Returns:: true if the objects are the same; false otherwise

exp

public static Complex exp(Complex z)

Returns the exponential of a Complex z, exp(z).

Specifically, if z = x+iy,
.
returns .
returns .
returns (where the signs of the real and imaginary parts of the result are unspecified).
returns (where the sign of the real part of the result is unspecified).
returns , for finite x.
returns , for finite y.
returns , for finite nonzero y.
returns (where the signs of the real and imaginary parts of the result are unspecified).
returns (where the sign of the real part of the result is unspecified).
returns .
returns , for all non-zero numbers y.
returns , for finite x.

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the exponential of the argument

floatValue

public float floatValue()

Returns the value of the real part as a float.

Specified by:: floatValue in class Number

Returns:: a float representing the value of the real part of a Complex object

hashCode

public int hashCode()

Returns a hashcode for this Complex.

Overrides:: hashCode in class Object

Returns:: a hash code value for this object

imag

public double imag()

Returns the imaginary part of a Complex object.

Returns:: a double representing the imaginary part of a Complex object, z

imag

public static double imag(Complex z)

Returns the imaginary part of a Complex object.

Parameters:: z - a Complex object
Returns:: a double representing the imaginary part of the Complex object, z

intValue

public int intValue()

Returns the value of the real part as an int.

Specified by:: intValue in class Number

Returns:: an int representing the value of the real part of a Complex object

log

public static Complex log(Complex z)

Returns the logarithm of a Complex z, with a branch cut along the negative real axis.

Specifically, if z = x+iy,
.
returns .
returns .
returns .
returns .
returns , for finite x.
returns , for finite positive-signed y.
returns , for finite positive-signed y.
returns .
returns .
returns , for finite x.
returns , for finite y.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to the logarithm of the argument. Its imaginary part is in the interval .

longValue

public long longValue()

Returns the value of the real part as a long.

Specified by:: longValue in class Number

Returns:: a long representing the value of the real part of a Complex object

multiply

public static Complex multiply(Complex x,
                               Complex y)

Returns the product of two Complex objects, x * y.

Parameters:: x - a Complex object; y - a Complex object
Returns:: a newly constructed Complex initialized to

multiply

public static Complex multiply(Complex x,
                               double y)

Returns the product of a Complex object and a double, x * y.

Parameters:: x - a Complex object; y - a double value
Returns:: a newly constructed Complex initialized to

multiply

public static Complex multiply(double x,
                               Complex y)

Returns the product of a double and a Complex object, x * y.

Parameters:: x - a double value; y - a Complex object
Returns:: a newly constructed Complex initialized to

multiplyImag

public static Complex multiplyImag(Complex x,
                                   double y)

Returns the product of a Complex object and a pure imaginary double, x * iy.

Parameters:: x - a Complex object; y - a double value representing a pure imaginary
Returns:: a newly constructed Complex initialized to x * iy

multiplyImag

public static Complex multiplyImag(double x,
                                   Complex y)

Returns the product of a pure imaginary double and a Complex object, ix * y.

Parameters:: x - a double value representing a pure imaginary; y - a Complex object
Returns:: a newly constructed Complex initialized to .

negate

public static Complex negate(Complex z)

Returns the negative of a Complex object, -z.

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the negative of the Complex argument, z

pow

public static Complex pow(Complex x,
                          Complex y)

Returns the Complex x raised to the Complex y power. The value of pow is defined in terms of the functions exp and log, by

Parameters:: x - a Complex object; y - a Complex object
Returns:: a newly constructed Complex initialized to .
See Also:: exp(com.imsl.math.Complex), log(com.imsl.math.Complex)

pow

public static Complex pow(Complex z,
                          double x)

Returns the Complex z raised to the x power, with a branch cut for the first parameter (z) along the negative real axis.

Parameters:: z - a Complex object; x - a double value
Returns:: a newly constructed Complex initialized to z to the power x

real

public double real()

Returns the real part of a Complex object.

Returns:: a double representing the real part of a Complex object, z

real

public static double real(Complex z)

Returns the real part of a Complex object.

Parameters:: z - a Complex object
Returns:: a double representing the real part of the Complex object, z

shortValue

public short shortValue()

Returns the value of the real part as a short.

Overrides:: shortValue in class Number

Returns:: a short representing the value of the real part of a Complex object

sin

public static Complex sin(Complex z)

Returns the sine of a Complex. The value of sin is defined in terms of the function sinh, by

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the sine of the argument
See Also:: sinh(com.imsl.math.Complex)

sinh

public static Complex sinh(Complex z)

Returns the hyperbolic sine of a Complex.

If z = x+iy,
and sinh is odd.
returns .
returns (where the sign of the real part of the result is unspecified).
returns .
returns (where the sign of the real part of the result is unspecified).
returns , for positive finite y.
returns , for positive finite x.
returns (where the sign of the real part of the result is unspecified).
returns (where the sign of the real part of the result is unspecified).
returns , for finite nonzero x.
returns .
returns , for all nonzero numbers y.
returns .

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the hyperbolic sine of the argument

sqrt

public static Complex sqrt(Complex z)

Returns the square root of a Complex, with a branch cut along the negative real axis.

Specifically, if z = x+iy,
.
returns .
returns , for finite positive-signed y.
returns , for finite positive-signed y.
returns , for all x (including NaN).
returns (where the sign of the imaginary part of the result is unspecified).
returns .
returns and optionally raises the invalid exception, for finite x.
returns and optionally raises the invalid exception, for finite y.
returns .

Parameters:: z - a Complex object
Returns:: A newly constructed Complex initialized to square root of z.

subtract

public static Complex subtract(Complex x,
                               Complex y)

Returns the difference of two Complex objects, x-y.

Parameters:: x - a Complex object; y - a Complex object
Returns:: a newly constructed Complex initialized to x-y

subtract

public static Complex subtract(Complex x,
                               double y)

Returns the difference of a Complex object and a double, x-y.

Parameters:: x - a Complex object; y - a double value
Returns:: a newly constructed Complex initialized to x-y

subtract

public static Complex subtract(double x,
                               Complex y)

Returns the difference of a double and a Complex object, x-y.

Parameters:: x - a double value; y - a Complex object
Returns:: a newly constructed Complex initialized to x-y

tan

public static Complex tan(Complex z)

Returns the tangent of a Complex. The value of tan is defined in terms of the function tanh, by

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the tangent of the argument
See Also:: tanh(com.imsl.math.Complex)

tanh

public static Complex tanh(Complex z)

Returns the hyperbolic tanh of a Complex.

If z = x+iy,
and tanh is odd.
returns .
returns , for all positive-signed numbers y.
returns , for finite x.
returns (where the sign of the imaginary part of the result is unspecified).
returns .
returns , for all nonzero numbers y.
returns , for finite x.
returns .

Parameters:: z - a Complex object
Returns:: a newly constructed Complex initialized to the hyperbolic tangent of the argument

toString

public String toString()

Returns a String representation for the specified Complex.

Overrides:: toString in class Object

Returns:: a String representation for this object

valueOf

public static Complex valueOf(String s)
                       throws NumberFormatException

Parses a String into a Complex.

Parameters:: s - the String to be parsed
Returns:: a newly constructed Complex initialized to the value represented by the String argument
Throws:: NumberFormatException - if the string does not contain a parsable Complex number; NullPointerException - if the input argument is null

Overview

Package

Class

Tree

Index

Help

JMSL^TM Numerical Library 6.1

PREV CLASS NEXT CLASS

FRAMES NO FRAMES

SUMMARY: NESTED | FIELD | CONSTR | METHOD

DETAIL: FIELD | CONSTR | METHOD

com.imsl.math Class Complex

i

suffix

Complex

Complex

Complex

Complex

abs

acos

acosh

add

add

add

argument

asin

asinh

atan

atanh

byteValue

compareTo

compareTo

conjugate

cos

cosh

divide

divide

divide

doubleValue

equals

equals

exp

floatValue

hashCode

imag

imag

intValue

log

longValue

multiply

multiply

multiply

multiplyImag

multiplyImag

negate

pow

pow

real

real

shortValue

sin

sinh

sqrt

subtract

subtract

subtract

tan

tanh

toString

valueOf

com.imsl.math
Class Complex