<?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()));
}
}
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3
Customer-Centric Approach
Your needs drive our solutions, and we are dedicated to delivering a superior user experience.
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Innovative Solutions
We continuously evolve, integrating the latest technologies to enhance your payment experience.
Testimonial Section
Customer Feedback: Real Experiences from Satisfied Clients and Partners
Hear from our users who trust NFC Pay for their everyday transactions. Our commitment to security, ease of use, and exceptional service shines through in their experiences. See why our clients choose NFC Pay for their payment needs and how it has transformed the way they manage their finances.
"NFC Pay has made my transactions incredibly simple and secure. The intuitive interface and quick payment options are game-changers for my business"
"I love how NFC Pay prioritizes security without compromising on convenience. The two-factor authentication and instant alerts give me peace of mind every time I use it."
"Setting up my merchant account was a breeze, and now I can accept payments effortlessly. NFC Pay has truly streamlined my operations, saving me time and hassle."
Get the NFC Pay App for Seamless Transactions Anytime, Anywhere
Unlock the full potential of NFC Pay by downloading our app, designed to bring secure, swift, and smart transactions to your fingertips. Whether you're paying at a store, transferring money to friends, or managing your business payments, the NFC Pay app makes it effortless. Available on both iOS and Android, it's your all-in-one solution for convenient and reliable digital payments. Download now and experience the future of payments!
In the digital age, privacy concerns have become increasingly paramount, prompting the European Union to enact the General Data Protection Regulation (GDPR) in 2018. Among its many provisions, GDPR sets strict guidelines for the collection and processing of personal data, including the use of cookies on websites. Privacy Policy
