ComplexUtils

Documentation of 'org.matheclipse.core.expression.ComplexUtils' Java class ComplexUtils

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java.lang.Object
- org.matheclipse.core.expression.ComplexUtils

```
public class ComplexUtils
extends java.lang.Object
```
Static implementations of common Complex-valued functions. Included are trigonometric, exponential, log, power and square root functions.
Reference:
- Multiple Precision DoubleComplexImpl Arithmetic and Functions
See individual method javadocs for the computational formulas used. In general, NaN values in either real or imaginary parts of input arguments result in ComplexNum.NaN returned. Otherwise, infinite or NaN values are returned as they arise in computing the real functions specified in the computational formulas. Null arguments result in NullPointerExceptions.

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static ComplexNum`	`acos(ComplexNum z)` Compute the inverse cosine for the given complex argument.
`static ComplexNum`	`asin(ComplexNum z)` Compute the inverse sine for the given complex argument.
`static ComplexNum`	`atan(ComplexNum z)` Compute the inverse tangent for the given complex argument.
`static ComplexNum`	`cos(ComplexNum z)` Compute the cosine for the given complex argument.
`static ComplexNum`	`cosh(ComplexNum z)` Compute the hyperbolic cosine for the given complex argument.
`static ComplexNum`	`exp(ComplexNum z)` Compute the exponential function for the given complex argument.
`static ComplexNum`	`log(ComplexNum z)` Compute the natural logarithm for the given complex argument.
`static ComplexNum`	`polar2Complex(double r, double theta)` Creates a complex number from the given polar representation.
`static ComplexNum`	`pow(ComplexNum y, ComplexNum x)` Returns of value of `y` raised to the power of `x`.
`static ComplexNum`	`sin(ComplexNum z)` Compute the sine for the given complex argument.
`static ComplexNum`	`sinh(ComplexNum z)` Compute the hyperbolic sine for the given complex argument.
`static ComplexNum`	`sqrt(ComplexNum z)` Compute the square root for the given complex argument.
`static ComplexNum`	`sqrt1z(ComplexNum z)` Compute the square root of 1 - `z`² for the given complex argument.
`static ComplexNum`	`tan(ComplexNum z)` Compute the tangent for the given complex argument.
`static ComplexNum`	`tanh(ComplexNum z)` Compute the hyperbolic tangent for the given complex argument.

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Method Detail
  - acos
```
public static ComplexNum acos(ComplexNum z)
```
    Compute the inverse cosine for the given complex argument.
    Implements the formula:
```
  acos(z) = -i (log(z + i (sqrt(1 - z²))))
```
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN or infinite.
    Parameters:
    
    z - the value whose inverse cosine is to be returned
    
    Returns:
    
    the inverse cosine of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - asin
```
public static ComplexNum asin(ComplexNum z)
```
    Compute the inverse sine for the given complex argument.
    Implements the formula:
```
  asin(z) = -i (log(sqrt(1 - z²) + iz)) 
```
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN or infinite.
    Parameters:
    
    z - the value whose inverse sine is to be returned.
    
    Returns:
    
    the inverse sine of z.
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - atan
```
public static ComplexNum atan(ComplexNum z)
```
    Compute the inverse tangent for the given complex argument.
    Implements the formula:
```
  atan(z) = (i/2) log((i + z)/(i - z)) 
```
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN or infinite.
    Parameters:
    
    z - the value whose inverse tangent is to be returned
    
    Returns:
    
    the inverse tangent of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - cos
```
public static ComplexNum cos(ComplexNum z)
```
    Compute the cosine for the given complex argument.
    Implements the formula:
```
  cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 cos(1 ± INFINITY i) = 1 ∓ INFINITY i
 cos(±INFINITY + i) = NaN + NaN i
 cos(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Parameters:
    
    z - the value whose cosine is to be returned
    
    Returns:
    
    the cosine of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - cosh
```
public static ComplexNum cosh(ComplexNum z)
```
    Compute the hyperbolic cosine for the given complex argument.
    Implements the formula:
```
  cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 cosh(1 ± INFINITY i) = NaN + NaN i
 cosh(±INFINITY + i) = INFINITY ± INFINITY i
 cosh(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Throws NullPointerException if z is null.
    Parameters:
    
    z - the value whose hyperbolic cosine is to be returned.
    
    Returns:
    
    the hyperbolic cosine of z.
  - exp
```
public static ComplexNum exp(ComplexNum z)
```
    Compute the exponential function for the given complex argument.
    Implements the formula:
```
  exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i
```
    where the (real) functions on the right-hand side are Math.exp(double), Math.cos(double), and Math.sin(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 exp(1 ± INFINITY i) = NaN + NaN i
 exp(INFINITY + i) = INFINITY + INFINITY i
 exp(-INFINITY + i) = 0 + 0i
 exp(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Throws NullPointerException if z is null.
    Parameters:
    
    z - the value
    
    Returns:
    
    e^z
  - log
```
public static ComplexNum log(ComplexNum z)
```
    Compute the natural logarithm for the given complex argument.
    Implements the formula:
```
  log(a + bi) = ln(|a + bi|) + arg(a + bi)i
```
    where ln on the right hand side is Math.log(double), |a + bi| is the modulus, ExprImpl.abs(), and arg(a + bi) = Math.atan2(double, double)(b, a)
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 log(1 ± INFINITY i) = INFINITY ± (π/2)i
 log(INFINITY + i) = INFINITY + 0i
 log(-INFINITY + i) = INFINITY + πi
 log(INFINITY ± INFINITY i) = INFINITY ± (π/4)i
 log(-INFINITY ± INFINITY i) = INFINITY ± (3π/4)i
 log(0 + 0i) = -INFINITY + 0i
 
```
    Throws NullPointerException if z is null.
    Parameters:
    
    z - the value.
    
    Returns:
    
    ln z.
  - polar2Complex
```
public static ComplexNum polar2Complex(double r,
                                       double theta)
```
    Creates a complex number from the given polar representation.
    The value returned is r·e^i·theta, computed as r·cos(theta) + r·sin(theta)i
    If either r or theta is NaN, or theta is infinite, ComplexNum.NaN is returned.
    If r is infinite and theta is finite, infinite or NaN values may be returned in parts of the result, following the rules for double arithmetic.
```
 Examples:
 
 polar2Complex(INFINITY, π/4) = INFINITY + INFINITY i
 polar2Complex(INFINITY, 0) = INFINITY + NaN i
 polar2Complex(INFINITY, -π/4) = INFINITY - INFINITY i
 polar2Complex(INFINITY, 5π/4) = -INFINITY - INFINITY i 
```
    Parameters:
    
    r - the modulus of the complex number to create
    
    theta - the argument of the complex number to create
    
    Returns:
    
    r·e^i·theta
    
    Throws:
    
    java.lang.IllegalArgumentException - if r is negative
    
    Since:
    
    1.1
  - pow
```
public static ComplexNum pow(ComplexNum y,
                             ComplexNum x)
```
    Returns of value of y raised to the power of x.
    Implements the formula:
```
  y^x = exp(x·log(y))
```
    where exp and log are exp(org.matheclipse.core.expression.ComplexNum) and log(org.matheclipse.core.expression.ComplexNum), respectively.
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN or infinite, or if y equals ComplexNum.ZERO.
    Parameters:
    
    y - the base.
    
    x - the exponent.
    
    Returns:
    
    y^x
    
    Throws:
    
    java.lang.NullPointerException - if either x or y is null
  - sin
```
public static ComplexNum sin(ComplexNum z)
```
    Compute the sine for the given complex argument.
    Implements the formula:
```
  sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sin(1 ± INFINITY i) = 1 ± INFINITY i
 sin(±INFINITY + i) = NaN + NaN i
 sin(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Throws NullPointerException if z is null.
    Parameters:
    
    z - the value whose sine is to be returned.
    
    Returns:
    
    the sine of z.
  - sinh
```
public static ComplexNum sinh(ComplexNum z)
```
    Compute the hyperbolic sine for the given complex argument.
    Implements the formula:
```
  sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sinh(1 ± INFINITY i) = NaN + NaN i
 sinh(±INFINITY + i) = ± INFINITY + INFINITY i
 sinh(±INFINITY ± INFINITY i) = NaN + NaN i

Parameters:
z - the value whose hyperbolic sine is to be returned
Returns:
the hyperbolic sine of z
Throws:
java.lang.NullPointerException - if z is null
```
  - sqrt
```
public static ComplexNum sqrt(ComplexNum z)
```
    Compute the square root for the given complex argument.
    Implements the following algorithm to compute sqrt(a + bi):
    1. Let t = sqrt((|a| + |a + bi|) / 2)
    2. if a ≥ 0 return t + (b/2t)i else return |b|/2t + sign(b)t i
    where
    - |a| = Math.abs(int)(a)
    - |a + bi| = ExprImpl.abs()(a + bi)
    - sign(b) = MathUtils.indicator(byte)(b)
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sqrt(1 ± INFINITY i) = INFINITY + NaN i
 sqrt(INFINITY + i) = INFINITY + 0i
 sqrt(-INFINITY + i) = 0 + INFINITY i
 sqrt(INFINITY ± INFINITY i) = INFINITY + NaN i
 sqrt(-INFINITY ± INFINITY i) = NaN ± INFINITY i
 
```
    Parameters:
    
    z - the value whose square root is to be returned
    
    Returns:
    
    the square root of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - sqrt1z
```
public static ComplexNum sqrt1z(ComplexNum z)
```
    Compute the square root of 1 - z² for the given complex argument.
    Computes the result directly as sqrt(DoubleComplexImpl.ONE.subtract(z.multiply(z))).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
    
    Parameters:
    
    z - the value
    
    Returns:
    
    the square root of 1 - z²
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - tan
```
public static ComplexNum tan(ComplexNum z)
```
    Compute the tangent for the given complex argument.
    Implements the formula:
```
 tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 tan(1 ± INFINITY i) = 0 + NaN i
 tan(±INFINITY + i) = NaN + NaN i
 tan(±INFINITY ± INFINITY i) = NaN + NaN i
 tan(±&pi/2 + 0 i) = ±INFINITY + NaN i
```
    Parameters:
    
    z - the value whose tangent is to be returned
    
    Returns:
    
    the tangent of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null
  - tanh
```
public static ComplexNum tanh(ComplexNum z)
```
    Compute the hyperbolic tangent for the given complex argument.
    Implements the formula:
```
 tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), MathUtils.cosh(double) and MathUtils.sinh(double).
    Returns ComplexNum.NaN if either real or imaginary part of the input argument is NaN.
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 tanh(1 ± INFINITY i) = NaN + NaN i
 tanh(±INFINITY + i) = NaN + 0 i
 tanh(±INFINITY ± INFINITY i) = NaN + NaN i
 tanh(0 + (&pi/2)i) = NaN + INFINITY i
```
    Parameters:
    
    z - the value whose hyperbolic tangent is to be returned
    
    Returns:
    
    the hyperbolic tangent of z
    
    Throws:
    
    java.lang.NullPointerException - if z is null

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Class ComplexUtils

Method Summary

Methods inherited from class java.lang.Object

Method Detail

acos

asin

atan

cos

cosh

exp

log

polar2Complex

pow

sin

sinh

sqrt

sqrt1z

tan

tanh