dev.nm.number.complex

Class ElementaryFunction

java.lang.Object

dev.nm.number.complex.ElementaryFunction

public class ElementaryFunction
extends 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

Constructors

Constructor and Description
ElementaryFunction()

Method Summary

Modifier and Type	Method and Description
static Complex	acos(Complex z) Inverse of cosine. \[ \arccos(z) = pi / 2 + i \ln(iz + \sqrt{1 - z^{2}}) = pi / 2 - \arcsin(z) \]
static Complex	asin(Complex z) Inverse of sine. \[ \arcsin(z) = -i \ln(iz + \sqrt{1 - z^{2}}) \]
static Complex	atan(Complex z) Inverse of tangent. \[ \arctan(z) = i \cdot ( \ln ( 1 - iz ) - \ln ( 1 + iz) ) / 2 \]
static Complex	cos(Complex z) Cosine of a complex number. \[ \cos(a + bi) = \cos(a)\cosh(b) - i \sin(a)\sinh(b) \]
static Complex	cosh(Complex z) Hyperbolic cosine of a complex number. \[ \cosh(a + bi) = \cosh(a)\cos(b) + i \sinh(a)\sin(b) \]
static Complex	exp(Complex z) Exponential of a complex number. \[ \begin{gathered} \exp(a + bi) = \exp(a) \cdot [\cos(b) + i \sin(b)] \\ = \exp(b)\cos(b) + i \exp(a)\sin(b) \end{gathered} \]
static Complex	log(Complex z) Natural logarithm of a complex number. \[ \ln(a + bi) = \ln(r \exp(i \theta )) = \ln(r) + i \theta \]
static Complex	pow(Complex z1, Complex z2) z1 to the power z2. \[ (r \exp(i \theta )) ^ (a + bi) = r^a \exp(-b \theta ) (\cos(b \ln(r) + a \theta ) + i \sin(b \ln(r) + a \theta )) \]
static Complex	sin(Complex z) Sine of a complex number. \[ \sin(a + bi) = \sin(a)\cosh(b) + i \cos(a)\sinh(b) \]
static Complex	sinh(Complex z) Hyperbolic sine of a complex number. \[ \sinh(a + bi) = \sinh(a)\cos(b) + i \cosh(a)\sin(b) \]
static Complex	sqrt(Complex z) Square root of a complex number.
static Complex	tan(Complex z) Tangent of a complex number. \[ \tan(a + bi) = [\sin(2a) + i \sinh(2a)] / [\cos(2a) + \cosh(2b)] \]
static Complex	tanh(Complex z) Hyperbolic tangent of a complex number. \[ \tanh(a + bi) = [\sinh(2a) + i \sin(2b)] / [\cosh(2a) + \cos(2b)] \]

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

Throws:

IllegalArgumentException - if z is a NaN or ∞

See Also:

Wikipedia: Square roots of negative and complex numbers

log

public static Complex log(Complex z)

Natural logarithm of a complex number. \[ \ln(a + bi) = \ln(r \exp(i \theta )) = \ln(r) + i \theta \]

Parameters:

z - a complex number

Returns:

ln(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

exp

public static Complex exp(Complex z)

Exponential of a complex number. \[ \begin{gathered} \exp(a + bi) = \exp(a) \cdot [\cos(b) + i \sin(b)] \\ = \exp(b)\cos(b) + i \exp(a)\sin(b) \end{gathered} \]

Parameters:

z - a complex number

Returns:

\(e^{z}\)

Throws:

IllegalArgumentException - if z is a NaN or ∞

pow

public static Complex pow(Complex z1,
                          Complex z2)

z₁ to the power z₂. \[ (r \exp(i \theta )) ^ (a + bi) = r^a \exp(-b \theta ) (\cos(b \ln(r) + a \theta ) + i \sin(b \ln(r) + a \theta )) \]

Parameters:

z1 - a complex number

z2 - a complex number

Returns:

\(z1^{z2}\)

Throws:

IllegalArgumentException - if z is a NaN or ∞

sin

public static Complex sin(Complex z)

Sine of a complex number. \[ \sin(a + bi) = \sin(a)\cosh(b) + i \cos(a)\sinh(b) \]

Parameters:

z - a complex number

Returns:

sin(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

cos

public static Complex cos(Complex z)

Cosine of a complex number. \[ \cos(a + bi) = \cos(a)\cosh(b) - i \sin(a)\sinh(b) \]

Parameters:

z - a complex number

Returns:

cos(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

tan

public static Complex tan(Complex z)

Tangent of a complex number. \[ \tan(a + bi) = [\sin(2a) + i \sinh(2a)] / [\cos(2a) + \cosh(2b)] \]

Parameters:

z - a complex number

Returns:

tan(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

asin

public static Complex asin(Complex z)

Inverse of sine. \[ \arcsin(z) = -i \ln(iz + \sqrt{1 - z^{2}}) \]

Parameters:

z - a complex number

Returns:

\(\sin^{-1}(z)\)

Throws:

IllegalArgumentException - if z is a NaN or ∞

See Also:

Inverse Sine from Wolfram MathWorld

acos

public static Complex acos(Complex z)

Inverse of cosine. \[ \arccos(z) = pi / 2 + i \ln(iz + \sqrt{1 - z^{2}}) = pi / 2 - \arcsin(z) \]

Parameters:

z - a complex number

Returns:

\(\cos^{-1}(z)\)

Throws:

IllegalArgumentException - if z is a NaN or ∞

See Also:

Inverse Cosine from Wolfram MathWorld

atan

public static Complex atan(Complex z)

Inverse of tangent. \[ \arctan(z) = i \cdot ( \ln ( 1 - iz ) - \ln ( 1 + iz) ) / 2 \]

Parameters:

z - a complex number

Returns:

\(\tan^{-1}(z)\)

Throws:

IllegalArgumentException - if z is a NaN or ∞

See Also:

Inverse Tangent from Wolfram MathWorld

sinh

public static Complex sinh(Complex z)

Hyperbolic sine of a complex number. \[ \sinh(a + bi) = \sinh(a)\cos(b) + i \cosh(a)\sin(b) \]

Parameters:

z - a complex number

Returns:

sinh(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

cosh

public static Complex cosh(Complex z)

Hyperbolic cosine of a complex number. \[ \cosh(a + bi) = \cosh(a)\cos(b) + i \sinh(a)\sin(b) \]

Parameters:

z - a complex number

Returns:

cosh(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞

tanh

public static Complex tanh(Complex z)

Hyperbolic tangent of a complex number. \[ \tanh(a + bi) = [\sinh(2a) + i \sin(2b)] / [\cosh(2a) + \cos(2b)] \]

Parameters:

z - a complex number

Returns:

tanh(z)

Throws:

IllegalArgumentException - if z is a NaN or ∞