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<?php namespace Complex; use InvalidArgumentException; class Functions { /** * Returns the absolute value (modulus) of a complex number. * Also known as the rho of the complex number, i.e. the distance/radius * from the centrepoint to the representation of the number in polar coordinates. * * This function is a synonym for rho() * * @param Complex|mixed $complex Complex number or a numeric value. * @return float The absolute (or rho) value of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * * @see rho * */ public static function abs($complex): float { return self::rho($complex); } /** * Returns the inverse cosine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse cosine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function acos($complex): Complex { $complex = Complex::validateComplexArgument($complex); $invsqrt = self::sqrt(Operations::subtract(1, Operations::multiply($complex, $complex))); $adjust = new Complex( $complex->getReal() - $invsqrt->getImaginary(), $complex->getImaginary() + $invsqrt->getReal() ); $log = self::ln($adjust); return new Complex( $log->getImaginary(), -1 * $log->getReal() ); } /** * Returns the inverse hyperbolic cosine of a complex number. * * Formula from Wolfram Alpha: * cosh^(-1)z = ln(z + sqrt(z + 1) sqrt(z - 1)). * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic cosine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function acosh($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal() && ($complex->getReal() > 1)) { return new Complex(\acosh($complex->getReal())); } $acosh = self::ln( Operations::add( $complex, Operations::multiply( self::sqrt(Operations::add($complex, 1)), self::sqrt(Operations::subtract($complex, 1)) ) ) ); return $acosh; } /** * Returns the inverse cotangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse cotangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function acot($complex): Complex { $complex = Complex::validateComplexArgument($complex); return self::atan(self::inverse($complex)); } /** * Returns the inverse hyperbolic cotangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic cotangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function acoth($complex): Complex { $complex = Complex::validateComplexArgument($complex); return self::atanh(self::inverse($complex)); } /** * Returns the inverse cosecant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse cosecant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function acsc($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::asin(self::inverse($complex)); } /** * Returns the inverse hyperbolic cosecant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic cosecant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function acsch($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::asinh(self::inverse($complex)); } /** * Returns the argument of a complex number. * Also known as the theta of the complex number, i.e. the angle in radians * from the real axis to the representation of the number in polar coordinates. * * This function is a synonym for theta() * * @param Complex|mixed $complex Complex number or a numeric value. * @return float The argument (or theta) value of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * * @see theta */ public static function argument($complex): float { return self::theta($complex); } /** * Returns the inverse secant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse secant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function asec($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::acos(self::inverse($complex)); } /** * Returns the inverse hyperbolic secant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic secant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function asech($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::acosh(self::inverse($complex)); } /** * Returns the inverse sine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse sine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function asin($complex): Complex { $complex = Complex::validateComplexArgument($complex); $invsqrt = self::sqrt(Operations::subtract(1, Operations::multiply($complex, $complex))); $adjust = new Complex( $invsqrt->getReal() - $complex->getImaginary(), $invsqrt->getImaginary() + $complex->getReal() ); $log = self::ln($adjust); return new Complex( $log->getImaginary(), -1 * $log->getReal() ); } /** * Returns the inverse hyperbolic sine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic sine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function asinh($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal() && ($complex->getReal() > 1)) { return new Complex(\asinh($complex->getReal())); } $asinh = clone $complex; $asinh = $asinh->reverse() ->invertReal(); $asinh = self::asin($asinh); return $asinh->reverse() ->invertImaginary(); } /** * Returns the inverse tangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse tangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function atan($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\atan($complex->getReal())); } $t1Value = new Complex(-1 * $complex->getImaginary(), $complex->getReal()); $uValue = new Complex(1, 0); $d1Value = clone $uValue; $d1Value = Operations::subtract($d1Value, $t1Value); $d2Value = Operations::add($t1Value, $uValue); $uResult = $d1Value->divideBy($d2Value); $uResult = self::ln($uResult); $realMultiplier = -0.5; $imaginaryMultiplier = 0.5; if (abs($uResult->getImaginary()) === M_PI) { // If we have an imaginary value at the max or min (PI or -PI), then we need to ensure // that the primary is assigned for the correct quadrant. $realMultiplier = ( ($uResult->getImaginary() === M_PI && $uResult->getReal() > 0.0) || ($uResult->getImaginary() === -M_PI && $uResult->getReal() < 0.0) ) ? 0.5 : -0.5; } return new Complex( $uResult->getImaginary() * $realMultiplier, $uResult->getReal() * $imaginaryMultiplier, $complex->getSuffix() ); } /** * Returns the inverse hyperbolic tangent of a complex number. * * Formula from Wolfram Alpha: * tanh^(-1)z = 1/2 [ln(1 + z) - ln(1 - z)]. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse hyperbolic tangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function atanh($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { $real = $complex->getReal(); if ($real >= -1.0 && $real <= 1.0) { return new Complex(\atanh($real)); } else { return new Complex(\atanh(1 / $real), (($real < 0.0) ? M_PI_2 : -1 * M_PI_2)); } } $atanh = Operations::multiply( Operations::subtract( self::ln(Operations::add(1.0, $complex)), self::ln(Operations::subtract(1.0, $complex)) ), 0.5 ); return $atanh; } /** * Returns the complex conjugate of a complex number * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The conjugate of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function conjugate($complex): Complex { $complex = Complex::validateComplexArgument($complex); return new Complex( $complex->getReal(), -1 * $complex->getImaginary(), $complex->getSuffix() ); } /** * Returns the cosine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The cosine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function cos($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\cos($complex->getReal())); } return self::conjugate( new Complex( \cos($complex->getReal()) * \cosh($complex->getImaginary()), \sin($complex->getReal()) * \sinh($complex->getImaginary()), $complex->getSuffix() ) ); } /** * Returns the hyperbolic cosine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic cosine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function cosh($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\cosh($complex->getReal())); } return new Complex( \cosh($complex->getReal()) * \cos($complex->getImaginary()), \sinh($complex->getReal()) * \sin($complex->getImaginary()), $complex->getSuffix() ); } /** * Returns the cotangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The cotangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function cot($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::inverse(self::tan($complex)); } /** * Returns the hyperbolic cotangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic cotangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function coth($complex): Complex { $complex = Complex::validateComplexArgument($complex); return self::inverse(self::tanh($complex)); } /** * Returns the cosecant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The cosecant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function csc($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::inverse(self::sin($complex)); } /** * Returns the hyperbolic cosecant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic cosecant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function csch($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { return new Complex(INF); } return self::inverse(self::sinh($complex)); } /** * Returns the exponential of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The exponential of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function exp($complex): Complex { $complex = Complex::validateComplexArgument($complex); if (($complex->getReal() == 0.0) && (\abs($complex->getImaginary()) == M_PI)) { return new Complex(-1.0, 0.0); } $rho = \exp($complex->getReal()); return new Complex( $rho * \cos($complex->getImaginary()), $rho * \sin($complex->getImaginary()), $complex->getSuffix() ); } /** * Returns the inverse of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The inverse of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws InvalidArgumentException If function would result in a division by zero */ public static function inverse($complex): Complex { $complex = clone Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0 && $complex->getImaginary() == 0.0) { throw new InvalidArgumentException('Division by zero'); } return $complex->divideInto(1.0); } /** * Returns the natural logarithm of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The natural logarithm of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws InvalidArgumentException If the real and the imaginary parts are both zero */ public static function ln($complex): Complex { $complex = Complex::validateComplexArgument($complex); if (($complex->getReal() == 0.0) && ($complex->getImaginary() == 0.0)) { throw new InvalidArgumentException(); } return new Complex( \log(self::rho($complex)), self::theta($complex), $complex->getSuffix() ); } /** * Returns the base-2 logarithm of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The base-2 logarithm of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws InvalidArgumentException If the real and the imaginary parts are both zero */ public static function log2($complex): Complex { $complex = Complex::validateComplexArgument($complex); if (($complex->getReal() == 0.0) && ($complex->getImaginary() == 0.0)) { throw new InvalidArgumentException(); } elseif (($complex->getReal() > 0.0) && ($complex->getImaginary() == 0.0)) { return new Complex(\log($complex->getReal(), 2), 0.0, $complex->getSuffix()); } return self::ln($complex) ->multiply(\log(Complex::EULER, 2)); } /** * Returns the common logarithm (base 10) of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The common logarithm (base 10) of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws InvalidArgumentException If the real and the imaginary parts are both zero */ public static function log10($complex): Complex { $complex = Complex::validateComplexArgument($complex); if (($complex->getReal() == 0.0) && ($complex->getImaginary() == 0.0)) { throw new InvalidArgumentException(); } elseif (($complex->getReal() > 0.0) && ($complex->getImaginary() == 0.0)) { return new Complex(\log10($complex->getReal()), 0.0, $complex->getSuffix()); } return self::ln($complex) ->multiply(\log10(Complex::EULER)); } /** * Returns the negative of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The negative value of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * * @see rho * */ public static function negative($complex): Complex { $complex = Complex::validateComplexArgument($complex); return new Complex( -1 * $complex->getReal(), -1 * $complex->getImaginary(), $complex->getSuffix() ); } /** * Returns a complex number raised to a power. * * @param Complex|mixed $complex Complex number or a numeric value. * @param float|integer $power The power to raise this value to * @return Complex The complex argument raised to the real power. * @throws Exception If the power argument isn't a valid real */ public static function pow($complex, $power): Complex { $complex = Complex::validateComplexArgument($complex); if (!is_numeric($power)) { throw new Exception('Power argument must be a real number'); } if ($complex->getImaginary() == 0.0 && $complex->getReal() >= 0.0) { return new Complex(\pow($complex->getReal(), $power)); } $rValue = \sqrt(($complex->getReal() * $complex->getReal()) + ($complex->getImaginary() * $complex->getImaginary())); $rPower = \pow($rValue, $power); $theta = $complex->argument() * $power; if ($theta == 0) { return new Complex(1); } return new Complex($rPower * \cos($theta), $rPower * \sin($theta), $complex->getSuffix()); } /** * Returns the rho of a complex number. * This is the distance/radius from the centrepoint to the representation of the number in polar coordinates. * * @param Complex|mixed $complex Complex number or a numeric value. * @return float The rho value of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function rho($complex): float { $complex = Complex::validateComplexArgument($complex); return \sqrt( ($complex->getReal() * $complex->getReal()) + ($complex->getImaginary() * $complex->getImaginary()) ); } /** * Returns the secant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The secant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function sec($complex): Complex { $complex = Complex::validateComplexArgument($complex); return self::inverse(self::cos($complex)); } /** * Returns the hyperbolic secant of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic secant of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function sech($complex): Complex { $complex = Complex::validateComplexArgument($complex); return self::inverse(self::cosh($complex)); } /** * Returns the sine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The sine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function sin($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\sin($complex->getReal())); } return new Complex( \sin($complex->getReal()) * \cosh($complex->getImaginary()), \cos($complex->getReal()) * \sinh($complex->getImaginary()), $complex->getSuffix() ); } /** * Returns the hyperbolic sine of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic sine of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function sinh($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\sinh($complex->getReal())); } return new Complex( \sinh($complex->getReal()) * \cos($complex->getImaginary()), \cosh($complex->getReal()) * \sin($complex->getImaginary()), $complex->getSuffix() ); } /** * Returns the square root of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The Square root of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function sqrt($complex): Complex { $complex = Complex::validateComplexArgument($complex); $theta = self::theta($complex); $delta1 = \cos($theta / 2); $delta2 = \sin($theta / 2); $rho = \sqrt(self::rho($complex)); return new Complex($delta1 * $rho, $delta2 * $rho, $complex->getSuffix()); } /** * Returns the tangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The tangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws InvalidArgumentException If function would result in a division by zero */ public static function tan($complex): Complex { $complex = Complex::validateComplexArgument($complex); if ($complex->isReal()) { return new Complex(\tan($complex->getReal())); } $real = $complex->getReal(); $imaginary = $complex->getImaginary(); $divisor = 1 + \pow(\tan($real), 2) * \pow(\tanh($imaginary), 2); if ($divisor == 0.0) { throw new InvalidArgumentException('Division by zero'); } return new Complex( \pow(self::sech($imaginary)->getReal(), 2) * \tan($real) / $divisor, \pow(self::sec($real)->getReal(), 2) * \tanh($imaginary) / $divisor, $complex->getSuffix() ); } /** * Returns the hyperbolic tangent of a complex number. * * @param Complex|mixed $complex Complex number or a numeric value. * @return Complex The hyperbolic tangent of the complex argument. * @throws Exception If argument isn't a valid real or complex number. * @throws \InvalidArgumentException If function would result in a division by zero */ public static function tanh($complex): Complex { $complex = Complex::validateComplexArgument($complex); $real = $complex->getReal(); $imaginary = $complex->getImaginary(); $divisor = \cos($imaginary) * \cos($imaginary) + \sinh($real) * \sinh($real); if ($divisor == 0.0) { throw new InvalidArgumentException('Division by zero'); } return new Complex( \sinh($real) * \cosh($real) / $divisor, 0.5 * \sin(2 * $imaginary) / $divisor, $complex->getSuffix() ); } /** * Returns the theta of a complex number. * This is the angle in radians from the real axis to the representation of the number in polar coordinates. * * @param Complex|mixed $complex Complex number or a numeric value. * @return float The theta value of the complex argument. * @throws Exception If argument isn't a valid real or complex number. */ public static function theta($complex): float { $complex = Complex::validateComplexArgument($complex); if ($complex->getReal() == 0.0) { if ($complex->isReal()) { return 0.0; } elseif ($complex->getImaginary() < 0.0) { return M_PI / -2; } return M_PI / 2; } elseif ($complex->getReal() > 0.0) { return \atan($complex->getImaginary() / $complex->getReal()); } elseif ($complex->getImaginary() < 0.0) { return -(M_PI - \atan(\abs($complex->getImaginary()) / \abs($complex->getReal()))); } return M_PI - \atan($complex->getImaginary() / \abs($complex->getReal())); } }