com.numericalmethod.suanshu.number.complex
Class ElementaryFunction

java.lang.Object
  com.numericalmethod.suanshu.number.complex.ElementaryFunction

public class ElementaryFunction
extends java.lang.Object

This class contains some elementary functions for complex number Complex. The references for the formulae can be found in the following.

See Also:

• Wikipedia: Square roots of negative and complex numbers
• Wikipedia: Logarithm
• Wikipedia: Exponentiation
• Wikipedia: Trigonometric functions
• Wikipedia: Hyperbolic functions

Constructor Summary

ElementaryFunction()

Method Summary

static Complex	acos(Complex z) Inverse of cosine.
static Complex	asin(Complex z) Inverse of sine.
static Complex	atan(Complex z) Inverse of tangent.
static Complex	cos(Complex z) Cosine of a complex number (a + bi).
static Complex	cosh(Complex z) Hyperbolic cosine of a complex number (a + bi).
static Complex	exp(Complex z) Exponential of a complex number (a + bi).
static Complex	log(Complex z) Natural logarithm of a complex number (a + bi).
static Complex	pow(Complex z1, Complex z2) z1 to the power z2.
static Complex	sin(Complex z) Sine of a complex number (a + bi).
static Complex	sinh(Complex z) Hyperbolic sine of a complex number (a + bi).
static Complex	sqrt(Complex z) Square root of a complex number.
static Complex	tan(Complex z) Tangent of a complex number (a + bi).
static Complex	tanh(Complex z) Hyperbolic tangent of a complex number (a + bi).

Methods inherited from class java.lang.Object

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

Constructor Detail

ElementaryFunction

public ElementaryFunction()

Method Detail

sqrt

public static Complex sqrt(Complex z)

Square root of a complex number.

Parameters:

z - a complex number

Returns:

the square root of the number, if not NaN of INFINITY

See Also:

Wikipedia: Square roots of negative and complex numbers

log

public static Complex log(Complex z)

Natural logarithm of a complex number (a + bi).

ln(a + bi) = ln(r exp(i θ)) = ln(r) + i θ

Parameters:

z - a complex number

Returns:

ln(z)

exp

public static Complex exp(Complex z)

Exponential of a complex number (a + bi).

 exp(a + bi) = exp(a) * [cos(b) + i sin(b)]
             = exp(b)cos(b) + i exp(a)sin(b)
 

Parameters:

z - a complex number

Returns:

ez

pow

public static Complex pow(Complex z1,
                          Complex z2)

z1 to the power z2.

(r exp(i θ)) ^ (a + bi) = r^a exp(-b θ) (cos(b ln(r) + a θ) + i sin(b ln(r) + a θ))

Parameters:

z1 - first complex number

z2 - second complex number

Returns:

z1z2

sin

public static Complex sin(Complex z)

Sine of a complex number (a + bi).

sin(a + bi) = sin(a)cosh(b) + i cos(a)sinh(b)

Parameters:

z - a complex number

Returns:

sin(z)

cos

public static Complex cos(Complex z)

Cosine of a complex number (a + bi).

cos(a + bi) = cos(a)cosh(b) - i sin(a)sinh(b)

Parameters:

z - a complex number

Returns:

cos(z)

tan

public static Complex tan(Complex z)

Tangent of a complex number (a + bi).

                sin(2a) + i sinh(2a)
 tan(a + bi) = ----------------------
                 cos(2a) + cosh(2b)
 

Parameters:

z - a complex number

Returns:

tan(z)

asin

public static Complex asin(Complex z)

Inverse of sine.

arcsin(z) = -i ln(iz + sqrt(1 - z2))

Parameters:

z - a complex number

Returns:

sin-1(z)

See Also:

Inverse Sine from Wolfram MathWorld

acos

public static Complex acos(Complex z)

Inverse of cosine.

arccos(z) = pi / 2 + i ln(iz + sqrt(1 - z2)) = pi / 2 - arcsin(z)

Parameters:

z - a complex number

Returns:

cos-1(z)

See Also:

Inverse Cosine from Wolfram MathWorld

atan

public static Complex atan(Complex z)

Inverse of tangent.

              i
 arctan(z) = --- ( ln ( 1 - iz ) - ln ( 1 + iz) )
              2
 

Parameters:

z - a complex number

Returns:

tan-1(z)

See Also:

Inverse Tangent from Wolfram MathWorld

sinh

public static Complex sinh(Complex z)

Hyperbolic sine of a complex number (a + bi).

sinh(a + bi) = sinh(a)cos(b) + i cosh(a)sin(b)

Parameters:

z - a complex number

Returns:

sinh(z)

cosh

public static Complex cosh(Complex z)

Hyperbolic cosine of a complex number (a + bi).

cosh(a + bi) = cosh(a)cos(b) + i sinh(a)sin(b)

Parameters:

z - a complex number

Returns:

cosh(z)

tanh

public static Complex tanh(Complex z)

Hyperbolic tangent of a complex number (a + bi).

                 sinh(2a) + i sin(2b)
 tanh(a + bi) = ----------------------
                  cosh(2a) + cos(2b)
 

Parameters:

z - a complex number

Returns:

tanh(z)