Java tutorial
/* * Licensed to the Apache Software Foundation (ASF) under one or more * contributor license agreements. See the NOTICE file distributed with * this work for additional information regarding copyright ownership. * The ASF licenses this file to You under the Apache License, Version 2.0 * (the "License"); you may not use this file except in compliance with * the License. You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ package org.apache.commons.math.complex; import java.io.Serializable; import java.util.ArrayList; import java.util.List; import org.apache.commons.math.FieldElement; import org.apache.commons.math.MathRuntimeException; import org.apache.commons.math.exception.util.LocalizedFormats; import org.apache.commons.math.util.MathUtils; import org.apache.commons.math.util.FastMath; /** * Representation of a Complex number - a number which has both a * real and imaginary part. * <p> * Implementations of arithmetic operations handle <code>NaN</code> and * infinite values according to the rules for {@link java.lang.Double} * arithmetic, applying definitional formulas and returning <code>NaN</code> or * infinite values in real or imaginary parts as these arise in computation. * See individual method javadocs for details.</p> * <p> * {@link #equals} identifies all values with <code>NaN</code> in either real * or imaginary part - e.g., <pre> * <code>1 + NaNi == NaN + i == NaN + NaNi.</code></pre></p> * * implements Serializable since 2.0 * * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 aot 2010) $ */ public class Complex implements FieldElement<Complex>, Serializable { /** The square root of -1. A number representing "0.0 + 1.0i" */ public static final Complex I = new Complex(0.0, 1.0); // CHECKSTYLE: stop ConstantName /** A complex number representing "NaN + NaNi" */ public static final Complex NaN = new Complex(Double.NaN, Double.NaN); // CHECKSTYLE: resume ConstantName /** A complex number representing "+INF + INFi" */ public static final Complex INF = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY); /** A complex number representing "1.0 + 0.0i" */ public static final Complex ONE = new Complex(1.0, 0.0); /** A complex number representing "0.0 + 0.0i" */ public static final Complex ZERO = new Complex(0.0, 0.0); /** Serializable version identifier */ private static final long serialVersionUID = -6195664516687396620L; /** The imaginary part. */ private final double imaginary; /** The real part. */ private final double real; /** Record whether this complex number is equal to NaN. */ private final transient boolean isNaN; /** Record whether this complex number is infinite. */ private final transient boolean isInfinite; /** * Create a complex number given the real and imaginary parts. * * @param real the real part * @param imaginary the imaginary part */ public Complex(double real, double imaginary) { super(); this.real = real; this.imaginary = imaginary; isNaN = Double.isNaN(real) || Double.isNaN(imaginary); isInfinite = !isNaN && (Double.isInfinite(real) || Double.isInfinite(imaginary)); } /** * Return the absolute value of this complex number. * <p> * Returns <code>NaN</code> if either real or imaginary part is * <code>NaN</code> and <code>Double.POSITIVE_INFINITY</code> if * neither part is <code>NaN</code>, but at least one part takes an infinite * value.</p> * * @return the absolute value */ public double abs() { if (isNaN()) { return Double.NaN; } if (isInfinite()) { return Double.POSITIVE_INFINITY; } if (FastMath.abs(real) < FastMath.abs(imaginary)) { if (imaginary == 0.0) { return FastMath.abs(real); } double q = real / imaginary; return FastMath.abs(imaginary) * FastMath.sqrt(1 + q * q); } else { if (real == 0.0) { return FastMath.abs(imaginary); } double q = imaginary / real; return FastMath.abs(real) * FastMath.sqrt(1 + q * q); } } /** * Return the sum of this complex number and the given complex number. * <p> * Uses the definitional formula * <pre> * (a + bi) + (c + di) = (a+c) + (b+d)i * </pre></p> * <p> * If either this or <code>rhs</code> has a NaN value in either part, * {@link #NaN} is returned; otherwise Inifinite and NaN values are * returned in the parts of the result according to the rules for * {@link java.lang.Double} arithmetic.</p> * * @param rhs the other complex number * @return the complex number sum * @throws NullPointerException if <code>rhs</code> is null */ public Complex add(Complex rhs) { return createComplex(real + rhs.getReal(), imaginary + rhs.getImaginary()); } /** * Return the conjugate of this complex number. The conjugate of * "A + Bi" is "A - Bi". * <p> * {@link #NaN} is returned if either the real or imaginary * part of this Complex number equals <code>Double.NaN</code>.</p> * <p> * If the imaginary part is infinite, and the real part is not NaN, * the returned value has infinite imaginary part of the opposite * sign - e.g. the conjugate of <code>1 + POSITIVE_INFINITY i</code> * is <code>1 - NEGATIVE_INFINITY i</code></p> * * @return the conjugate of this Complex object */ public Complex conjugate() { if (isNaN()) { return NaN; } return createComplex(real, -imaginary); } /** * Return the quotient of this complex number and the given complex number. * <p> * Implements the definitional formula * <pre><code> * a + bi ac + bd + (bc - ad)i * ----------- = ------------------------- * c + di c<sup>2</sup> + d<sup>2</sup> * </code></pre> * but uses * <a href="http://doi.acm.org/10.1145/1039813.1039814"> * prescaling of operands</a> to limit the effects of overflows and * underflows in the computation.</p> * <p> * Infinite and NaN values are handled / returned according to the * following rules, applied in the order presented: * <ul> * <li>If either this or <code>rhs</code> has a NaN value in either part, * {@link #NaN} is returned.</li> * <li>If <code>rhs</code> equals {@link #ZERO}, {@link #NaN} is returned. * </li> * <li>If this and <code>rhs</code> are both infinite, * {@link #NaN} is returned.</li> * <li>If this is finite (i.e., has no infinite or NaN parts) and * <code>rhs</code> is infinite (one or both parts infinite), * {@link #ZERO} is returned.</li> * <li>If this is infinite and <code>rhs</code> is finite, NaN values are * returned in the parts of the result if the {@link java.lang.Double} * rules applied to the definitional formula force NaN results.</li> * </ul></p> * * @param rhs the other complex number * @return the complex number quotient * @throws NullPointerException if <code>rhs</code> is null */ public Complex divide(Complex rhs) { if (isNaN() || rhs.isNaN()) { return NaN; } double c = rhs.getReal(); double d = rhs.getImaginary(); if (c == 0.0 && d == 0.0) { return NaN; } if (rhs.isInfinite() && !isInfinite()) { return ZERO; } if (FastMath.abs(c) < FastMath.abs(d)) { double q = c / d; double denominator = c * q + d; return createComplex((real * q + imaginary) / denominator, (imaginary * q - real) / denominator); } else { double q = d / c; double denominator = d * q + c; return createComplex((imaginary * q + real) / denominator, (imaginary - real * q) / denominator); } } /** * Test for the equality of two Complex objects. * <p> * If both the real and imaginary parts of two Complex numbers * are exactly the same, and neither is <code>Double.NaN</code>, the two * Complex objects are considered to be equal.</p> * <p> * All <code>NaN</code> values are considered to be equal - i.e, if either * (or both) real and imaginary parts of the complex number are equal * to <code>Double.NaN</code>, the complex number is equal to * <code>Complex.NaN</code>.</p> * * @param other Object to test for equality to this * @return true if two Complex objects are equal, false if * object is null, not an instance of Complex, or * not equal to this Complex instance * */ @Override public boolean equals(Object other) { if (this == other) { return true; } if (other instanceof Complex) { Complex rhs = (Complex) other; if (rhs.isNaN()) { return this.isNaN(); } else { return (real == rhs.real) && (imaginary == rhs.imaginary); } } return false; } /** * Get a hashCode for the complex number. * <p> * All NaN values have the same hash code.</p> * * @return a hash code value for this object */ @Override public int hashCode() { if (isNaN()) { return 7; } return 37 * (17 * MathUtils.hash(imaginary) + MathUtils.hash(real)); } /** * Access the imaginary part. * * @return the imaginary part */ public double getImaginary() { return imaginary; } /** * Access the real part. * * @return the real part */ public double getReal() { return real; } /** * Returns true if either or both parts of this complex number is NaN; * false otherwise * * @return true if either or both parts of this complex number is NaN; * false otherwise */ public boolean isNaN() { return isNaN; } /** * Returns true if either the real or imaginary part of this complex number * takes an infinite value (either <code>Double.POSITIVE_INFINITY</code> or * <code>Double.NEGATIVE_INFINITY</code>) and neither part * is <code>NaN</code>. * * @return true if one or both parts of this complex number are infinite * and neither part is <code>NaN</code> */ public boolean isInfinite() { return isInfinite; } /** * Return the product of this complex number and the given complex number. * <p> * Implements preliminary checks for NaN and infinity followed by * the definitional formula: * <pre><code> * (a + bi)(c + di) = (ac - bd) + (ad + bc)i * </code></pre> * </p> * <p> * Returns {@link #NaN} if either this or <code>rhs</code> has one or more * NaN parts. * </p> * Returns {@link #INF} if neither this nor <code>rhs</code> has one or more * NaN parts and if either this or <code>rhs</code> has one or more * infinite parts (same result is returned regardless of the sign of the * components). * </p> * <p> * Returns finite values in components of the result per the * definitional formula in all remaining cases. * </p> * * @param rhs the other complex number * @return the complex number product * @throws NullPointerException if <code>rhs</code> is null */ public Complex multiply(Complex rhs) { if (isNaN() || rhs.isNaN()) { return NaN; } if (Double.isInfinite(real) || Double.isInfinite(imaginary) || Double.isInfinite(rhs.real) || Double.isInfinite(rhs.imaginary)) { // we don't use Complex.isInfinite() to avoid testing for NaN again return INF; } return createComplex(real * rhs.real - imaginary * rhs.imaginary, real * rhs.imaginary + imaginary * rhs.real); } /** * Return the product of this complex number and the given scalar number. * <p> * Implements preliminary checks for NaN and infinity followed by * the definitional formula: * <pre><code> * c(a + bi) = (ca) + (cb)i * </code></pre> * </p> * <p> * Returns {@link #NaN} if either this or <code>rhs</code> has one or more * NaN parts. * </p> * Returns {@link #INF} if neither this nor <code>rhs</code> has one or more * NaN parts and if either this or <code>rhs</code> has one or more * infinite parts (same result is returned regardless of the sign of the * components). * </p> * <p> * Returns finite values in components of the result per the * definitional formula in all remaining cases. * </p> * * @param rhs the scalar number * @return the complex number product */ public Complex multiply(double rhs) { if (isNaN() || Double.isNaN(rhs)) { return NaN; } if (Double.isInfinite(real) || Double.isInfinite(imaginary) || Double.isInfinite(rhs)) { // we don't use Complex.isInfinite() to avoid testing for NaN again return INF; } return createComplex(real * rhs, imaginary * rhs); } /** * Return the additive inverse of this complex number. * <p> * Returns <code>Complex.NaN</code> if either real or imaginary * part of this Complex number equals <code>Double.NaN</code>.</p> * * @return the negation of this complex number */ public Complex negate() { if (isNaN()) { return NaN; } return createComplex(-real, -imaginary); } /** * Return the difference between this complex number and the given complex * number. * <p> * Uses the definitional formula * <pre> * (a + bi) - (c + di) = (a-c) + (b-d)i * </pre></p> * <p> * If either this or <code>rhs</code> has a NaN value in either part, * {@link #NaN} is returned; otherwise inifinite and NaN values are * returned in the parts of the result according to the rules for * {@link java.lang.Double} arithmetic. </p> * * @param rhs the other complex number * @return the complex number difference * @throws NullPointerException if <code>rhs</code> is null */ public Complex subtract(Complex rhs) { if (isNaN() || rhs.isNaN()) { return NaN; } return createComplex(real - rhs.getReal(), imaginary - rhs.getImaginary()); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseCosine.html" TARGET="_top"> * inverse cosine</a> of this complex number. * <p> * Implements the formula: <pre> * <code> acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>))))</code></pre></p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code> or infinite.</p> * * @return the inverse cosine of this complex number * @since 1.2 */ public Complex acos() { if (isNaN()) { return Complex.NaN; } return this.add(this.sqrt1z().multiply(Complex.I)).log().multiply(Complex.I.negate()); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseSine.html" TARGET="_top"> * inverse sine</a> of this complex number. * <p> * Implements the formula: <pre> * <code> asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz)) </code></pre></p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code> or infinite.</p> * * @return the inverse sine of this complex number. * @since 1.2 */ public Complex asin() { if (isNaN()) { return Complex.NaN; } return sqrt1z().add(this.multiply(Complex.I)).log().multiply(Complex.I.negate()); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseTangent.html" TARGET="_top"> * inverse tangent</a> of this complex number. * <p> * Implements the formula: <pre> * <code> atan(z) = (i/2) log((i + z)/(i - z)) </code></pre></p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code> or infinite.</p> * * @return the inverse tangent of this complex number * @since 1.2 */ public Complex atan() { if (isNaN()) { return Complex.NaN; } return this.add(Complex.I).divide(Complex.I.subtract(this)).log() .multiply(Complex.I.divide(createComplex(2.0, 0.0))); } /** * Compute the * <a href="http://mathworld.wolfram.com/Cosine.html" TARGET="_top"> * cosine</a> * of this complex number. * <p> * Implements the formula: <pre> * <code> cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * cos(1 ± INFINITY i) = 1 ∓ INFINITY i * cos(±INFINITY + i) = NaN + NaN i * cos(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> * * @return the cosine of this complex number * @since 1.2 */ public Complex cos() { if (isNaN()) { return Complex.NaN; } return createComplex(FastMath.cos(real) * MathUtils.cosh(imaginary), -FastMath.sin(real) * MathUtils.sinh(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/HyperbolicCosine.html" TARGET="_top"> * hyperbolic cosine</a> of this complex number. * <p> * Implements the formula: <pre> * <code> cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * cosh(1 ± INFINITY i) = NaN + NaN i * cosh(±INFINITY + i) = INFINITY ± INFINITY i * cosh(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> * * @return the hyperbolic cosine of this complex number. * @since 1.2 */ public Complex cosh() { if (isNaN()) { return Complex.NaN; } return createComplex(MathUtils.cosh(real) * FastMath.cos(imaginary), MathUtils.sinh(real) * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/ExponentialFunction.html" TARGET="_top"> * exponential function</a> of this complex number. * <p> * Implements the formula: <pre> * <code> exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#exp}, {@link java.lang.Math#cos}, and * {@link java.lang.Math#sin}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * 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</code></pre></p> * * @return <i>e</i><sup><code>this</code></sup> * @since 1.2 */ public Complex exp() { if (isNaN()) { return Complex.NaN; } double expReal = FastMath.exp(real); return createComplex(expReal * FastMath.cos(imaginary), expReal * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/NaturalLogarithm.html" TARGET="_top"> * natural logarithm</a> of this complex number. * <p> * Implements the formula: <pre> * <code> log(a + bi) = ln(|a + bi|) + arg(a + bi)i</code></pre> * where ln on the right hand side is {@link java.lang.Math#log}, * <code>|a + bi|</code> is the modulus, {@link Complex#abs}, and * <code>arg(a + bi) = {@link java.lang.Math#atan2}(b, a)</code></p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * 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.<pre> * Examples: * <code> * 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 * </code></pre></p> * * @return ln of this complex number. * @since 1.2 */ public Complex log() { if (isNaN()) { return Complex.NaN; } return createComplex(FastMath.log(abs()), FastMath.atan2(imaginary, real)); } /** * Returns of value of this complex number raised to the power of <code>x</code>. * <p> * Implements the formula: <pre> * <code> y<sup>x</sup> = exp(x·log(y))</code></pre> * where <code>exp</code> and <code>log</code> are {@link #exp} and * {@link #log}, respectively.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code> or infinite, or if <code>y</code> * equals {@link Complex#ZERO}.</p> * * @param x the exponent. * @return <code>this</code><sup><code>x</code></sup> * @throws NullPointerException if x is null * @since 1.2 */ public Complex pow(Complex x) { if (x == null) { throw new NullPointerException(); } return this.log().multiply(x).exp(); } /** * Compute the * <a href="http://mathworld.wolfram.com/Sine.html" TARGET="_top"> * sine</a> * of this complex number. * <p> * Implements the formula: <pre> * <code> sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * sin(1 ± INFINITY i) = 1 ± INFINITY i * sin(±INFINITY + i) = NaN + NaN i * sin(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> * * @return the sine of this complex number. * @since 1.2 */ public Complex sin() { if (isNaN()) { return Complex.NaN; } return createComplex(FastMath.sin(real) * MathUtils.cosh(imaginary), FastMath.cos(real) * MathUtils.sinh(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/HyperbolicSine.html" TARGET="_top"> * hyperbolic sine</a> of this complex number. * <p> * Implements the formula: <pre> * <code> sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * sinh(1 ± INFINITY i) = NaN + NaN i * sinh(±INFINITY + i) = ± INFINITY + INFINITY i * sinh(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> * * @return the hyperbolic sine of this complex number * @since 1.2 */ public Complex sinh() { if (isNaN()) { return Complex.NaN; } return createComplex(MathUtils.sinh(real) * FastMath.cos(imaginary), MathUtils.cosh(real) * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top"> * square root</a> of this complex number. * <p> * Implements the following algorithm to compute <code>sqrt(a + bi)</code>: * <ol><li>Let <code>t = sqrt((|a| + |a + bi|) / 2)</code></li> * <li><pre>if <code> a ≥ 0</code> return <code>t + (b/2t)i</code> * else return <code>|b|/2t + sign(b)t i </code></pre></li> * </ol> * where <ul> * <li><code>|a| = {@link Math#abs}(a)</code></li> * <li><code>|a + bi| = {@link Complex#abs}(a + bi) </code></li> * <li><code>sign(b) = {@link MathUtils#indicator}(b) </code> * </ul></p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * 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 * </code></pre></p> * * @return the square root of this complex number * @since 1.2 */ public Complex sqrt() { if (isNaN()) { return Complex.NaN; } if (real == 0.0 && imaginary == 0.0) { return createComplex(0.0, 0.0); } double t = FastMath.sqrt((FastMath.abs(real) + abs()) / 2.0); if (real >= 0.0) { return createComplex(t, imaginary / (2.0 * t)); } else { return createComplex(FastMath.abs(imaginary) / (2.0 * t), MathUtils.indicator(imaginary) * t); } } /** * Compute the * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top"> * square root</a> of 1 - <code>this</code><sup>2</sup> for this complex * number. * <p> * Computes the result directly as * <code>sqrt(Complex.ONE.subtract(z.multiply(z)))</code>.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.</p> * * @return the square root of 1 - <code>this</code><sup>2</sup> * @since 1.2 */ public Complex sqrt1z() { return createComplex(1.0, 0.0).subtract(this.multiply(this)).sqrt(); } /** * Compute the * <a href="http://mathworld.wolfram.com/Tangent.html" TARGET="_top"> * tangent</a> of this complex number. * <p> * Implements the formula: <pre> * <code>tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * 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.<pre> * Examples: * <code> * tan(1 ± INFINITY i) = 0 + NaN i * tan(±INFINITY + i) = NaN + NaN i * tan(±INFINITY ± INFINITY i) = NaN + NaN i * tan(±π/2 + 0 i) = ±INFINITY + NaN i</code></pre></p> * * @return the tangent of this complex number * @since 1.2 */ public Complex tan() { if (isNaN()) { return Complex.NaN; } double real2 = 2.0 * real; double imaginary2 = 2.0 * imaginary; double d = FastMath.cos(real2) + MathUtils.cosh(imaginary2); return createComplex(FastMath.sin(real2) / d, MathUtils.sinh(imaginary2) / d); } /** * Compute the * <a href="http://mathworld.wolfram.com/HyperbolicTangent.html" TARGET="_top"> * hyperbolic tangent</a> of this complex number. * <p> * Implements the formula: <pre> * <code>tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i</code></pre> * where the (real) functions on the right-hand side are * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is <code>NaN</code>.</p> * <p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.<pre> * Examples: * <code> * tanh(1 ± INFINITY i) = NaN + NaN i * tanh(±INFINITY + i) = NaN + 0 i * tanh(±INFINITY ± INFINITY i) = NaN + NaN i * tanh(0 + (π/2)i) = NaN + INFINITY i</code></pre></p> * * @return the hyperbolic tangent of this complex number * @since 1.2 */ public Complex tanh() { if (isNaN()) { return Complex.NaN; } double real2 = 2.0 * real; double imaginary2 = 2.0 * imaginary; double d = MathUtils.cosh(real2) + FastMath.cos(imaginary2); return createComplex(MathUtils.sinh(real2) / d, FastMath.sin(imaginary2) / d); } /** * <p>Compute the argument of this complex number. * </p> * <p>The argument is the angle phi between the positive real axis and the point * representing this number in the complex plane. The value returned is between -PI (not inclusive) * and PI (inclusive), with negative values returned for numbers with negative imaginary parts. * </p> * <p>If either real or imaginary part (or both) is NaN, NaN is returned. Infinite parts are handled * as java.Math.atan2 handles them, essentially treating finite parts as zero in the presence of * an infinite coordinate and returning a multiple of pi/4 depending on the signs of the infinite * parts. See the javadoc for java.Math.atan2 for full details.</p> * * @return the argument of this complex number */ public double getArgument() { return FastMath.atan2(getImaginary(), getReal()); } /** * <p>Computes the n-th roots of this complex number. * </p> * <p>The nth roots are defined by the formula: <pre> * <code> z<sub>k</sub> = abs<sup> 1/n</sup> (cos(phi + 2πk/n) + i (sin(phi + 2πk/n))</code></pre> * for <i><code>k=0, 1, ..., n-1</code></i>, where <code>abs</code> and <code>phi</code> are * respectively the {@link #abs() modulus} and {@link #getArgument() argument} of this complex number. * </p> * <p>If one or both parts of this complex number is NaN, a list with just one element, * {@link #NaN} is returned.</p> * <p>if neither part is NaN, but at least one part is infinite, the result is a one-element * list containing {@link #INF}.</p> * * @param n degree of root * @return List<Complex> all nth roots of this complex number * @throws IllegalArgumentException if parameter n is less than or equal to 0 * @since 2.0 */ public List<Complex> nthRoot(int n) throws IllegalArgumentException { if (n <= 0) { throw MathRuntimeException .createIllegalArgumentException(LocalizedFormats.CANNOT_COMPUTE_NTH_ROOT_FOR_NEGATIVE_N, n); } List<Complex> result = new ArrayList<Complex>(); if (isNaN()) { result.add(Complex.NaN); return result; } if (isInfinite()) { result.add(Complex.INF); return result; } // nth root of abs -- faster / more accurate to use a solver here? final double nthRootOfAbs = FastMath.pow(abs(), 1.0 / n); // Compute nth roots of complex number with k = 0, 1, ... n-1 final double nthPhi = getArgument() / n; final double slice = 2 * FastMath.PI / n; double innerPart = nthPhi; for (int k = 0; k < n; k++) { // inner part final double realPart = nthRootOfAbs * FastMath.cos(innerPart); final double imaginaryPart = nthRootOfAbs * FastMath.sin(innerPart); result.add(createComplex(realPart, imaginaryPart)); innerPart += slice; } return result; } /** * Create a complex number given the real and imaginary parts. * * @param realPart the real part * @param imaginaryPart the imaginary part * @return a new complex number instance * @since 1.2 */ protected Complex createComplex(double realPart, double imaginaryPart) { return new Complex(realPart, imaginaryPart); } /** * <p>Resolve the transient fields in a deserialized Complex Object.</p> * <p>Subclasses will need to override {@link #createComplex} to deserialize properly</p> * @return A Complex instance with all fields resolved. * @since 2.0 */ protected final Object readResolve() { return createComplex(real, imaginary); } /** {@inheritDoc} */ public ComplexField getField() { return ComplexField.getInstance(); } }