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funciones<complex>

abs

Calcula el módulo de un número complejo.

template <class Type>
Type abs(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo módulo se va a determinar.

Valor devuelto

El módulo de un número complejo.

Comentarios

El módulo de un número complejo es una medida de la longitud del vector que representa el número complejo. El módulo de un número complejo a + bi es la raíz cuadrada de (a2 + b2), escrita | a + bi|. La norma de un número complejo a + bi es (a2 + b2). La norma de un número complejo es el cuadrado de su módulo.

Ejemplo

// complex_abs.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // Complex numbers can be entered in polar form with
   // modulus and argument parameter inputs but are
   // stored in Cartesian form as real & imag coordinates
   complex <double> c1 ( polar ( 5.0 ) );   // Default argument = 0
   complex <double> c2 ( polar ( 5.0 , pi / 6 ) );
   complex <double> c3 ( polar ( 5.0 , 13 * pi / 6 ) );
   cout << "c1 = polar ( 5.0 ) = " << c1 << endl;
   cout << "c2 = polar ( 5.0 , pi / 6 ) = " << c2 << endl;
   cout << "c3 = polar ( 5.0 , 13 * pi / 6 ) = " << c3 << endl;

   // The modulus and argument of a complex number can be recovered
   // using abs & arg member functions
   double absc1 = abs ( c1 );
   double argc1 = arg ( c1 );
   cout << "The modulus of c1 is recovered from c1 using: abs ( c1 ) = "
        << absc1 << endl;
   cout << "Argument of c1 is recovered from c1 using:\n arg ( c1 ) = "
        << argc1 << " radians, which is " << argc1 * 180 / pi
        << " degrees." << endl;

   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is recovered from c2 using: abs ( c2 ) = "
        << absc2 << endl;
   cout << "Argument of c2 is recovered from c2 using:\n arg ( c2 ) = "
        << argc2 << " radians, which is " << argc2 * 180 / pi
        << " degrees." << endl;

   // Testing if the principal angles of c2 and c3 are the same
   if ( (arg ( c2 ) <= ( arg ( c3 ) + .00000001) ) ||
        (arg ( c2 ) >= ( arg ( c3 ) - .00000001) ) )
      cout << "The complex numbers c2 & c3 have the "
           << "same principal arguments."<< endl;
   else
      cout << "The complex numbers c2 & c3 don't have the "
           << "same principal arguments." << endl;
}
c1 = polar ( 5.0 ) = (5,0)
c2 = polar ( 5.0 , pi / 6 ) = (4.33013,2.5)
c3 = polar ( 5.0 , 13 * pi / 6 ) = (4.33013,2.5)
The modulus of c1 is recovered from c1 using: abs ( c1 ) = 5
Argument of c1 is recovered from c1 using:
arg ( c1 ) = 0 radians, which is 0 degrees.
The modulus of c2 is recovered from c2 using: abs ( c2 ) = 5
Argument of c2 is recovered from c2 using:
arg ( c2 ) = 0.523599 radians, which is 30 degrees.
The complex numbers c2 & c3 have the same principal arguments.

acos

template<class T> complex<T> acos(const complex<T>&);

acosh

template<class T> complex<T> acosh(const complex<T>&);

argumentos

Extrae el argumento de un número complejo.

template <class Type>
Type arg(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo argumento se va a determinar.

Valor devuelto

El argumento del número complejo.

Comentarios

El argumento es el ángulo que forma el vector complejo con el eje real positivo en el plano complejo. Para un número complejo a + bi, el argumento es igual a arctan( b/a). El ángulo tiene sentido positivo cuando se mide en dirección opuesta a las agujas del reloj desde el eje real positivo y un sentido negativo cuando se mide en la dirección de las agujas del reloj. Los valores principales son mayores que –pi y menores o iguales que +pi.

Ejemplo

// complex_arg.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // Complex numbers can be entered in polar form with
   // modulus and argument parameter inputs but are
   // stored in Cartesian form as real & imag coordinates
   complex <double> c1 ( polar ( 5.0 ) );   // Default argument = 0
   complex <double> c2 ( polar ( 5.0 , pi / 6 ) );
   complex <double> c3 ( polar ( 5.0 , 13 * pi / 6 ) );
   cout << "c1 = polar ( 5.0 ) = " << c1 << endl;
   cout << "c2 = polar ( 5.0 , pi / 6 ) = " << c2 << endl;
   cout << "c3 = polar ( 5.0 , 13 * pi / 6 ) = " << c3 << endl;

   // The modulus and argument of a complex number can be rcovered
   // using abs & arg member functions
   double absc1 = abs ( c1 );
   double argc1 = arg ( c1 );
   cout << "The modulus of c1 is recovered from c1 using: abs ( c1 ) = "
        << absc1 << endl;
   cout << "Argument of c1 is recovered from c1 using:\n arg ( c1 ) = "
        << argc1 << " radians, which is " << argc1 * 180 / pi
        << " degrees." << endl;

   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is recovered from c2 using: abs ( c2 ) = "
        << absc2 << endl;
   cout << "Argument of c2 is recovered from c2 using:\n arg ( c2 ) = "
        << argc2 << " radians, which is " << argc2 * 180 / pi
        << " degrees." << endl;

   // Testing if the principal angles of c2 and c3 are the same
   if ( (arg ( c2 ) <= ( arg ( c3 ) + .00000001) ) ||
        (arg ( c2 ) >= ( arg ( c3 ) - .00000001) ) )
      cout << "The complex numbers c2 & c3 have the "
           << "same principal arguments."<< endl;
   else
      cout << "The complex numbers c2 & c3 don't have the "
           << "same principal arguments." << endl;
}
c1 = polar ( 5.0 ) = (5,0)
c2 = polar ( 5.0 , pi / 6 ) = (4.33013,2.5)
c3 = polar ( 5.0 , 13 * pi / 6 ) = (4.33013,2.5)
The modulus of c1 is recovered from c1 using: abs ( c1 ) = 5
Argument of c1 is recovered from c1 using:
arg ( c1 ) = 0 radians, which is 0 degrees.
The modulus of c2 is recovered from c2 using: abs ( c2 ) = 5
Argument of c2 is recovered from c2 using:
arg ( c2 ) = 0.523599 radians, which is 30 degrees.
The complex numbers c2 & c3 have the same principal arguments.

asin

template<class T> complex<T> asin(const complex<T>&);

asinh

template<class T> complex<T> asinh(const complex<T>&);

atan

template<class T> complex<T> atan(const complex<T>&);

atanh

template<class T> complex<T> atanh(const complex<T>&);

conj

Devuelve el conjugado complejo de un número complejo.

template <class Type>
complex<Type> conj(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo conjugado complejo se va a devolver.

Valor devuelto

El conjugado complejo del número complejo de entrada.

Comentarios

El conjugado complejo de un número complejo a + bi es a – bi. El producto de un número complejo y su conjugado es la norma del número a2 + b2.

Ejemplo

// complex_conj.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;

   complex <double> c1 ( 4.0 , 3.0 );
   cout << "The complex number c1 = " << c1 << endl;

   double dr1 = real ( c1 );
   cout << "The real part of c1 is real ( c1 ) = "
        << dr1 << "." << endl;

   double di1 = imag ( c1 );
   cout << "The imaginary part of c1 is imag ( c1 ) = "
        << di1 << "." << endl;

   complex <double> c2 = conj ( c1 );
   cout << "The complex conjugate of c1 is c2 = conj ( c1 )= "
        << c2 << endl;

   double dr2 = real ( c2 );
   cout << "The real part of c2 is real ( c2 ) = "
        << dr2 << "." << endl;

   double di2 = imag ( c2 );
   cout << "The imaginary part of c2 is imag ( c2 ) = "
        << di2 << "." << endl;

   // The real part of the product of a complex number
   // and its conjugate is the norm of the number
   complex <double> c3 = c1 * c2;
   cout << "The norm of (c1 * conj (c1) ) is c1 * c2 = "
        << real( c3 ) << endl;
}
The complex number c1 = (4,3)
The real part of c1 is real ( c1 ) = 4.
The imaginary part of c1 is imag ( c1 ) = 3.
The complex conjugate of c1 is c2 = conj ( c1 )= (4,-3)
The real part of c2 is real ( c2 ) = 4.
The imaginary part of c2 is imag ( c2 ) = -3.
The norm of (c1 * conj (c1) ) is c1 * c2 = 25

cos

Devuelve el coseno de un número complejo.

template <class Type>
complex<Type> cos(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo coseno se está determinando.

Valor devuelto

El número complejo que es el coseno del número complejo de entrada.

Comentarios

Identidades que definen los cosenos complejos:

cos (z) = (1/2)*(exp (iz) + exp (- iz) )

cos (z) = cos (a + bi) = cos (a) cosh ( b) - isin (a) sinh (b)

Ejemplo

// complex_cos.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of cosine of a complex number c1
   complex <double> c2 = cos ( c1 );
   cout << "Complex number c2 = cos ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Cosines of the standard angles in the first
   // two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar (1.0, pi / 6) );
   v1.push_back( cos ( vc1 ) );
   complex <double> vc2  ( polar (1.0, pi / 3) );
   v1.push_back( cos ( vc2 ) );
   complex <double> vc3  ( polar (1.0, pi / 2) );
   v1.push_back( cos ( vc3) );
   complex <double> vc4  ( polar (1.0, 2 * pi / 3) );
   v1.push_back( cos ( vc4 ) );
   complex <double> vc5  ( polar (1.0, 5 * pi / 6) );
   v1.push_back( cos ( vc5 ) );
   complex <double> vc6  ( polar (1.0,  pi ) );
   v1.push_back( cos ( vc6 ) );

   cout << "The complex components cos (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = cos ( c1 ) = (-27.0349,-3.85115)
The modulus of c2 is: 27.3079
The argument of c2 is: -3.00009 radians, which is -171.893 degrees.

The complex components cos (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(0.730543,-0.39695)
(1.22777,-0.469075)
(1.54308,1.21529e-013)
(1.22777,0.469075)
(0.730543,0.39695)
(0.540302,-1.74036e-013)

cosh

Devuelve el coseno hiperbólico de un número complejo.

template <class Type>
complex<Type> cosh(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo coseno hiperbólico se está determinando.

Valor devuelto

El número complejo que es el coseno hiperbólico del número complejo de entrada.

Comentarios

Identidades que definen los cosenos hiperbólicos complejos:

cos (z) = (1/2)*( exp (z) + exp (- z) )

cos (z) = cosh (a + bi) = cosh (a) cos (b) + isinh (a) sin (b)

Ejemplo

// complex_cosh.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of cosine of a complex number c1
   complex <double> c2 = cosh ( c1 );
   cout << "Complex number c2 = cosh ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Hyperbolic cosines of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar (1.0, pi / 6) );
   v1.push_back( cosh ( vc1 ) );
   complex <double> vc2  ( polar (1.0, pi / 3) );
   v1.push_back( cosh ( vc2 ) );
   complex <double> vc3  ( polar (1.0, pi / 2) );
   v1.push_back( cosh ( vc3) );
   complex <double> vc4  ( polar (1.0, 2 * pi / 3) );
   v1.push_back( cosh ( vc4 ) );
   complex <double> vc5  ( polar (1.0, 5 * pi / 6) );
   v1.push_back( cosh ( vc5 ) );
   complex <double> vc6  ( polar (1.0,  pi ) );
   v1.push_back( cosh ( vc6 ) );

   cout << "The complex components cosh (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = cosh ( c1 ) = (-6.58066,-7.58155)
The modulus of c2 is: 10.0392
The argument of c2 is: -2.28564 radians, which is -130.957 degrees.

The complex components cosh (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(1.22777,0.469075)
(0.730543,0.39695)
(0.540302,-8.70178e-014)
(0.730543,-0.39695)
(1.22777,-0.469075)
(1.54308,2.43059e-013)

exp

Devuelve la función exponencial de un número complejo.

template <class Type>
complex<Type> exp(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo exponencial se está determinando.

Valor devuelto

El número complejo que es el exponencial del número complejo de entrada.

Ejemplo

// complex_exp.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main() {
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 1 , pi/6 );
   cout << "Complex number c1 = " << c1 << endl;

   // Value of exponential of a complex number c1:
   // note the argument of c2 is determined by the
   // imaginary part of c1 & the modulus by the real part
   complex <double> c2 = exp ( c1 );
   cout << "Complex number c2 = exp ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Exponentials of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( 0.0 , -pi );
   v1.push_back( exp ( vc1 ) );
   complex <double> vc2  ( 0.0, -2 * pi / 3 );
   v1.push_back( exp ( vc2 ) );
   complex <double> vc3  ( 0.0, 0.0 );
   v1.push_back( exp ( vc3 ) );
   complex <double> vc4  ( 0.0, pi / 3 );
   v1.push_back( exp ( vc4 ) );
   complex <double> vc5  ( 0.0 , 2 * pi / 3 );
   v1.push_back( exp ( vc5 ) );
   complex <double> vc6  ( 0.0, pi );
   v1.push_back( exp ( vc6 ) );

   cout << "The complex components exp (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 3 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin() ; Iter1 != v1.end() ; Iter1++ )
      cout <<  ( * Iter1 ) << "\n     with argument = "
           << ( 180/pi ) * arg ( *Iter1 )
           << " degrees\n     modulus = "
           << abs ( * Iter1 ) << endl;
}

imag

Extrae el componente imaginario de un número complejo.

template <class Type>
Type imag(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya parte real va a extraerse.

Valor devuelto

La parte imaginaria del número complejo como una función global.

Comentarios

Esta función de muestra no puede usarse para modificar la parte real del número complejo. Para cambiar la parte real, el valor del componente debe asignarse a un nuevo número complejo.

Ejemplo

// complexc_imag.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   complex <double> c1 ( 4.0 , 3.0 );
   cout << "The complex number c1 = " << c1 << endl;

   double dr1 = real ( c1 );
   cout << "The real part of c1 is real ( c1 ) = "
        << dr1 << "." << endl;

   double di1 = imag ( c1 );
   cout << "The imaginary part of c1 is imag ( c1 ) = "
        << di1 << "." << endl;
}
The complex number c1 = (4,3)
The real part of c1 is real ( c1 ) = 4.
The imaginary part of c1 is imag ( c1 ) = 3.

log

Devuelve el logaritmo natural de un número complejo.

template <class Type>
complex<Type> log(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo logaritmo natural se está determinando.

Valor devuelto

El número complejo que es el logaritmo natural del número complejo de entrada.

Comentarios

Los cortes de rama están en el eje real negativo.

Ejemplo

// complex_log.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main() {
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of log of a complex number c1
   complex <double> c2 = log ( c1 );
   cout << "Complex number c2 = log ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // log of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar (1.0, pi / 6) );
   v1.push_back( log ( vc1 ) );
   complex <double> vc2  ( polar (1.0, pi / 3) );
   v1.push_back( log ( vc2 ) );
   complex <double> vc3  ( polar (1.0, pi / 2) );
   v1.push_back( log ( vc3) );
   complex <double> vc4  ( polar (1.0, 2 * pi / 3) );
   v1.push_back( log ( vc4 ) );
   complex <double> vc5  ( polar (1.0, 5 * pi / 6) );
   v1.push_back( log ( vc5 ) );
   complex <double> vc6  ( polar (1.0,  pi ) );
   v1.push_back( log ( vc6 ) );

   cout << "The complex components log (vci), where abs (vci) = 1 "
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin() ; Iter1 != v1.end() ; Iter1++ )
      cout << *Iter1 << " " << endl;
}

log10

Devuelve el logaritmo de base 10 de un número complejo.

template <class Type>
complex<Type> log10(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo logaritmo de base 10 se está determinando.

Valor devuelto

El número complejo que es el logaritmo de base 10 del número complejo de entrada.

Comentarios

Los cortes de rama están en el eje real negativo.

Ejemplo

// complex_log10.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main() {
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of log10 of a complex number c1
   complex <double> c2 = log10 ( c1 );
   cout << "Complex number c2 = log10 ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // log10 of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar (1.0, pi / 6) );
   v1.push_back( log10 ( vc1 ) );
   complex <double> vc2  ( polar (1.0, pi / 3) );
   v1.push_back( log10 ( vc2 ) );
   complex <double> vc3  ( polar (1.0, pi / 2) );
   v1.push_back( log10 ( vc3) );
   complex <double> vc4  ( polar (1.0, 2 * pi / 3) );
   v1.push_back( log10 ( vc4 ) );
   complex <double> vc5  ( polar (1.0, 5 * pi / 6) );
   v1.push_back( log10 ( vc5 ) );
   complex <double> vc6  ( polar (1.0,  pi ) );
   v1.push_back( log10 ( vc6 ) );

   cout << "The complex components log10 (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}

norm

Extrae la norma de un número complejo.

template <class Type>
Type norm(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya norma se va a determinar.

Valor devuelto

La norma de un número complejo.

Comentarios

La norma de un número complejo a + bi es (a2 + b2). La norma de un número complejo es el cuadrado de su módulo. El módulo de un número complejo es una medida de la longitud del vector que representa el número complejo. El módulo de un número complejo a + bi es la raíz cuadrada de (a2 + b2), escrita | a + bi|.

Ejemplo

// complex_norm.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // Complex numbers can be entered in polar form with
   // modulus and argument parameter inputs but are
   // stored in Cartesian form as real & imag coordinates
   complex <double> c1 ( polar ( 5.0 ) );   // Default argument = 0
   complex <double> c2 ( polar ( 5.0 , pi / 6 ) );
   complex <double> c3 ( polar ( 5.0 , 13 * pi / 6 ) );
   cout << "c1 = polar ( 5.0 ) = " << c1 << endl;
   cout << "c2 = polar ( 5.0 , pi / 6 ) = " << c2 << endl;
   cout << "c3 = polar ( 5.0 , 13 * pi / 6 ) = " << c3 << endl;

   if ( (arg ( c2 ) <= ( arg ( c3 ) + .00000001) ) ||
        (arg ( c2 ) >= ( arg ( c3 ) - .00000001) ) )
      cout << "The complex numbers c2 & c3 have the "
           << "same principal arguments."<< endl;
   else
      cout << "The complex numbers c2 & c3 don't have the "
           << "same principal arguments." << endl;

   // The modulus and argument of a complex number can be recovered
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is recovered from c2 using: abs ( c2 ) = "
        << absc2 << endl;
   cout << "Argument of c2 is recovered from c2 using:\n arg ( c2 ) = "
        << argc2 << " radians, which is " << argc2 * 180 / pi
        << " degrees." << endl;

   // The norm of a complex number is the square of its modulus
   double normc2 = norm ( c2 );
   double sqrtnormc2 = sqrt ( normc2 );
   cout << "The norm of c2 given by: norm ( c2 ) = " << normc2 << endl;
   cout << "The modulus of c2 is the square root of the norm: "
        << "sqrt ( normc2 ) = " << sqrtnormc2 << ".";
}
c1 = polar ( 5.0 ) = (5,0)
c2 = polar ( 5.0 , pi / 6 ) = (4.33013,2.5)
c3 = polar ( 5.0 , 13 * pi / 6 ) = (4.33013,2.5)
The complex numbers c2 & c3 have the same principal arguments.
The modulus of c2 is recovered from c2 using: abs ( c2 ) = 5
Argument of c2 is recovered from c2 using:
arg ( c2 ) = 0.523599 radians, which is 30 degrees.
The norm of c2 given by: norm ( c2 ) = 25
The modulus of c2 is the square root of the norm: sqrt ( normc2 ) = 5.

polar

Devuelve el número complejo, que corresponde a un módulo y argumento especificados, en formato cartesiano.

template <class Type>
complex<Type> polar(const Type& _Modulus, const Type& _Argument = 0);

Parámetros

_Modulus
El módulo del número complejo que se va a introducir.

_Argument
El argumento del número complejo que se va a introducir.

Valor devuelto

Formato cartesiano del número complejo especificado en el formato polar.

Comentarios

La forma polar de un número complejo proporciona el módulo r y el argumento p, donde estos parámetros están relacionados con los componentes cartesianos reales e imaginarios a y b mediante las ecuaciones a = r * cos p y b = r * sin p.

Ejemplo

// complex_polar.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // Complex numbers can be entered in polar form with
   // modulus and argument parameter inputs but are
   // stored in Cartesian form as real & imag coordinates
   complex <double> c1 ( polar ( 5.0 ) );   // Default argument = 0
   complex <double> c2 ( polar ( 5.0 , pi / 6 ) );
   complex <double> c3 ( polar ( 5.0 , 13 * pi / 6 ) );
   cout << "c1 = polar ( 5.0 ) = " << c1 << endl;
   cout << "c2 = polar ( 5.0 , pi / 6 ) = " << c2 << endl;
   cout << "c3 = polar ( 5.0 , 13 * pi / 6 ) = " << c3 << endl;

   if ( (arg ( c2 ) <= ( arg ( c3 ) + .00000001) ) ||
        (arg ( c2 ) >= ( arg ( c3 ) - .00000001) ) )
      cout << "The complex numbers c2 & c3 have the "
           << "same principal arguments."<< endl;
   else
      cout << "The complex numbers c2 & c3 don't have the "
           << "same principal arguments." << endl;

   // the modulus and argument of a complex number can be rcovered
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is recovered from c2 using: abs ( c2 ) = "
        << absc2 << endl;
   cout << "Argument of c2 is recovered from c2 using:\n arg ( c2 ) = "
        << argc2 << " radians, which is " << argc2 * 180 / pi
        << " degrees." << endl;
}
c1 = polar ( 5.0 ) = (5,0)
c2 = polar ( 5.0 , pi / 6 ) = (4.33013,2.5)
c3 = polar ( 5.0 , 13 * pi / 6 ) = (4.33013,2.5)
The complex numbers c2 & c3 have the same principal arguments.
The modulus of c2 is recovered from c2 using: abs ( c2 ) = 5
Argument of c2 is recovered from c2 using:
arg ( c2 ) = 0.523599 radians, which is 30 degrees.

pow

Evalúa el número complejo obtenido al elevar una base que es un número complejo a la potencia de otro número complejo.

template <class Type>
complex<Type> pow(const complex<Type>& _Base, int _Power);

template <class Type>
complex<Type> pow(const complex<Type>& _Base, const Type& _Power);

template <class Type>
complex<Type> pow(const complex<Type>& _Base, const complex<Type>& _Power);

template <class Type>
complex<Type> pow(const Type& _Base, const complex<Type>& _Power);

Parámetros

_Base
El número complejo o un número que es del tipo de parámetro para el número complejo que es la base que se elevará a una potencia mediante la función miembro.

_Power
El número entero o complejo o un número que es del tipo de parámetro para el número complejo que es la potencia a la que se elevará la base mediante la función miembro.

Valor devuelto

El número complejo obtenido al elevar la base especificada a la potencia especificada.

Comentarios

Las funciones convierten de forma eficaz los dos operandos al tipo de valor devuelto y después devuelven el izquierdo convertido a la potencia derecha.

El corte de rama está en el eje real negativo.

Ejemplo

// complex_pow.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // First member function
   // type complex<double> base & type integer power
   complex <double> cb1 ( 3 , 4);
   int cp1 = 2;
   complex <double> ce1 = pow ( cb1 ,cp1 );

   cout << "Complex number for base cb1 = " << cb1 << endl;
   cout << "Integer for power = " << cp1 << endl;
   cout << "Complex number returned from complex base and integer power:"
        << "\n ce1 = cb1 ^ cp1 = " << ce1 << endl;
   double absce1 = abs ( ce1 );
   double argce1 = arg ( ce1 );
   cout << "The modulus of ce1 is: " << absce1 << endl;
   cout << "The argument of ce1 is: "<< argce1 << " radians, which is "
        << argce1 * 180 / pi << " degrees." << endl << endl;

   // Second member function
   // type complex<double> base & type double power
   complex <double> cb2 ( 3 , 4 );
   double cp2 = pi;
   complex <double> ce2 = pow ( cb2 ,cp2 );

   cout << "Complex number for base cb2 = " << cb2 << endl;
   cout << "Type double for power cp2 = pi = " << cp2 << endl;
   cout << "Complex number returned from complex base and double power:"
        << "\n ce2 = cb2 ^ cp2 = " << ce2 << endl;
   double absce2 = abs ( ce2 );
   double argce2 = arg ( ce2 );
   cout << "The modulus of ce2 is: " << absce2 << endl;
   cout << "The argument of ce2 is: "<< argce2 << " radians, which is "
        << argce2 * 180 / pi << " degrees." << endl << endl;

   // Third member function
   // type complex<double> base & type complex<double> power
   complex <double> cb3 ( 3 , 4 );
   complex <double> cp3 ( -2 , 1 );
   complex <double> ce3 = pow ( cb3 ,cp3 );

   cout << "Complex number for base cb3 = " << cb3 << endl;
   cout << "Complex number for power cp3= " << cp3 << endl;
   cout << "Complex number returned from complex base and complex power:"
        << "\n ce3 = cb3 ^ cp3 = " << ce3 << endl;
   double absce3 = abs ( ce3 );
   double argce3 = arg ( ce3 );
   cout << "The modulus of ce3 is: " << absce3 << endl;
   cout << "The argument of ce3 is: "<< argce3 << " radians, which is "
        << argce3 * 180 / pi << " degrees." << endl << endl;

   // Fourth member function
   // type double base & type complex<double> power
   double cb4 = pi;
   complex <double> cp4 ( 2 , -1 );
   complex <double> ce4 = pow ( cb4 ,cp4 );

   cout << "Type double for base cb4 = pi = " << cb4 << endl;
   cout << "Complex number for power cp4 = " << cp4 << endl;
   cout << "Complex number returned from double base and complex power:"
        << "\n ce4 = cb4 ^ cp4 = " << ce4 << endl;
   double absce4 = abs ( ce4 );
   double argce4 = arg ( ce4 );
   cout << "The modulus of ce4 is: " << absce4 << endl;
   cout << "The argument of ce4 is: "<< argce4 << " radians, which is "
        << argce4 * 180 / pi << " degrees." << endl << endl;
}
Complex number for base cb1 = (3,4)
Integer for power = 2
Complex number returned from complex base and integer power:
ce1 = cb1 ^ cp1 = (-7,24)
The modulus of ce1 is: 25
The argument of ce1 is: 1.85459 radians, which is 106.26 degrees.

Complex number for base cb2 = (3,4)
Type double for power cp2 = pi = 3.14159
Complex number returned from complex base and double power:
ce2 = cb2 ^ cp2 = (-152.915,35.5475)
The modulus of ce2 is: 156.993
The argument of ce2 is: 2.91318 radians, which is 166.913 degrees.

Complex number for base cb3 = (3,4)
Complex number for power cp3= (-2,1)
Complex number returned from complex base and complex power:
ce3 = cb3 ^ cp3 = (0.0153517,-0.00384077)
The modulus of ce3 is: 0.0158249
The argument of ce3 is: -0.245153 radians, which is -14.0462 degrees.

Type double for base cb4 = pi = 3.14159
Complex number for power cp4 = (2,-1)
Complex number returned from double base and complex power:
ce4 = cb4 ^ cp4 = (4.07903,-8.98725)
The modulus of ce4 is: 9.8696
The argument of ce4 is: -1.14473 radians, which is -65.5882 degrees.

proj

template<class T> complex<T> proj(const complex<T>&);

real

Extrae el componente real de un número complejo

template <class Type>
Type real(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya parte real va a extraerse.

Valor devuelto

La parte real del número complejo como función global.

Comentarios

Esta función de muestra no puede usarse para modificar la parte real del número complejo. Para cambiar la parte real, el valor del componente debe asignarse a un nuevo número complejo.

Ejemplo

// complex_real.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   complex <double> c1 ( 4.0 , 3.0 );
   cout << "The complex number c1 = " << c1 << endl;

   double dr1 = real ( c1 );
   cout << "The real part of c1 is real ( c1 ) = "
        << dr1 << "." << endl;

   double di1 = imag ( c1 );
   cout << "The imaginary part of c1 is imag ( c1 ) = "
        << di1 << "." << endl;
}
The complex number c1 = (4,3)
The real part of c1 is real ( c1 ) = 4.
The imaginary part of c1 is imag ( c1 ) = 3.

sin

Devuelve el seno de un número complejo.

template <class Type>
complex<Type> sin(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo seno se está determinando.

Valor devuelto

El número complejo que es el seno del número complejo de entrada.

Comentarios

Identidades que definen los senos complejos:

sin (z) = (1/2 i)*( exp (iz) - exp (- iz) )

sin (z) = sin (a + bi) = sin (a) cosh (b) + icos (a) sinh (b)

Ejemplo

// complex_sin.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of sine of a complex number c1
   complex <double> c2 = sin ( c1 );
   cout << "Complex number c2 = sin ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // sines of the standard angles in the first
   // two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar ( 1.0, pi / 6 ) );
   v1.push_back( sin ( vc1 ) );
   complex <double> vc2  ( polar ( 1.0, pi / 3 ) );
   v1.push_back( sin ( vc2 ) );
   complex <double> vc3  ( polar ( 1.0, pi / 2 ) );
   v1.push_back( sin ( vc3 ) );
   complex <double> vc4  ( polar ( 1.0, 2 * pi / 3 ) );
   v1.push_back( sin ( vc4 ) );
   complex <double> vc5  ( polar ( 1.0, 5 * pi / 6 ) );
   v1.push_back( sin ( vc5 ) );
   complex <double> vc6  ( polar ( 1.0, pi ) );
   v1.push_back( sin ( vc6 ) );

   cout << "The complex components sin (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = sin ( c1 ) = (3.85374,-27.0168)
The modulus of c2 is: 27.2903
The argument of c2 is: -1.42911 radians, which is -81.882 degrees.

The complex components sin (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(0.85898,0.337596)
(0.670731,0.858637)
(-1.59572e-013,1.1752)
(-0.670731,0.858637)
(-0.85898,0.337596)
(-0.841471,-1.11747e-013)

sinh

Devuelve el seno hiperbólico de un número complejo.

template <class Type>
complex<Type> sinh(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuyo seno hiperbólico se está determinando.

Valor devuelto

El número complejo que es el seno hiperbólico del número complejo de entrada.

Comentarios

Identidades que definen los senos hiperbólicos complejos:

sinh (z) = (1/2)*( exp (z) - exp (- z) )

sinh (z) = sinh (a + bi) = sinh (a) cos (b) + icosh (a) sin (b)

Ejemplo

// complex_sinh.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of sine of a complex number c1
   complex <double> c2 = sinh ( c1 );
   cout << "Complex number c2 = sinh ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Hyperbolic sines of the standard angles in
   // the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar ( 1.0, pi / 6 ) );
   v1.push_back( sinh ( vc1 ) );
   complex <double> vc2  ( polar ( 1.0, pi / 3 ) );
   v1.push_back( sinh ( vc2 ) );
   complex <double> vc3  ( polar ( 1.0, pi / 2 ) );
   v1.push_back( sinh ( vc3) );
   complex <double> vc4  ( polar ( 1.0, 2 * pi / 3 ) );
   v1.push_back( sinh ( vc4 ) );
   complex <double> vc5  ( polar ( 1.0, 5 * pi / 6 ) );
   v1.push_back( sinh ( vc5 ) );
   complex <double> vc6  ( polar ( 1.0, pi ) );
   v1.push_back( sinh ( vc6 ) );

   cout << "The complex components sinh (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = sinh ( c1 ) = (-6.54812,-7.61923)
The modulus of c2 is: 10.0464
The argument of c2 is: -2.28073 radians, which is -130.676 degrees.

The complex components sinh (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(0.858637,0.670731)
(0.337596,0.85898)
(-5.58735e-014,0.841471)
(-0.337596,0.85898)
(-0.858637,0.670731)
(-1.1752,-3.19145e-013)

sqrt

Calcula la raíz cuadrada de un número complejo.

template <class Type>
complex<Type> sqrt(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya raíz cuadrada se va a calcular.

Valor devuelto

La raíz cuadrada de un número complejo.

Comentarios

La raíz cuadrada tendrá un ángulo de fase en el intervalo semiabierto (-pi/2 y pi/2].

Los cortes de rama del plano complejo están en el eje real negativo.

La raíz cuadrada de un número complejo tendrá un módulo que es la raíz cuadrada del número de entrada y un argumento que es la mitad que el número de entrada.

Ejemplo

// complex_sqrt.cpp
// compile with: /EHsc
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;

   // Complex numbers can be entered in polar form with
   // modulus and argument parameter inputs but are
   // stored in Cartesian form as real & imag coordinates
   complex <double> c1 ( polar ( 25.0 , pi / 2 ) );
   complex <double> c2 = sqrt ( c1 );
   cout << "c1 = polar ( 5.0 ) = " << c1 << endl;
   cout << "c2 = sqrt ( c1 ) = " << c2 << endl;

   // The modulus and argument of a complex number can be recovered
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is recovered from c2 using: abs ( c2 ) = "
        << absc2 << endl;
   cout << "Argument of c2 is recovered from c2 using:\n arg ( c2 ) = "
        << argc2 << " radians, which is " << argc2 * 180 / pi
        << " degrees." << endl;

   // The modulus and argument of c2 can be directly calculated
   absc2 = sqrt( abs ( c1 ) );
   argc2 = 0.5 * arg ( c1 );
   cout << "The modulus of c2 = sqrt( abs ( c1 ) ) =" << absc2 << endl;
   cout << "The argument of c2 = ( 1 / 2 ) * arg ( c1 ) ="
        << argc2 << " radians,\n which is " << argc2 * 180 / pi
        << " degrees." << endl;
}
c1 = polar ( 5.0 ) = (-2.58529e-012,25)
c2 = sqrt ( c1 ) = (3.53553,3.53553)
The modulus of c2 is recovered from c2 using: abs ( c2 ) = 5
Argument of c2 is recovered from c2 using:
arg ( c2 ) = 0.785398 radians, which is 45 degrees.
The modulus of c2 = sqrt( abs ( c1 ) ) =5
The argument of c2 = ( 1 / 2 ) * arg ( c1 ) =0.785398 radians,
which is 45 degrees.

tan

Devuelve la tangente de un número complejo.

template <class Type>
complex<Type> tan(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya tangente se está determinando.

Valor devuelto

El número complejo que es la tangente del número complejo de entrada.

Comentarios

Identidades que definen la cotangente compleja:

tan (z) = sin (z) / cos (z) = ( exp (iz) - exp (- iz) ) / i( exp (iz) + exp (- iz) )

Ejemplo

// complex_tan.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of cosine of a complex number c1
   complex <double> c2 = tan ( c1 );
   cout << "Complex number c2 = tan ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Hyperbolic tangent of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar ( 1.0, pi / 6 ) );
   v1.push_back( tan ( vc1 ) );
   complex <double> vc2  ( polar ( 1.0, pi / 3 ) );
   v1.push_back( tan ( vc2 ) );
   complex <double> vc3  ( polar ( 1.0, pi / 2 ) );
   v1.push_back( tan ( vc3) );
   complex <double> vc4  ( polar ( 1.0, 2 * pi / 3 ) );
   v1.push_back( tan ( vc4 ) );
   complex <double> vc5  ( polar ( 1.0, 5 * pi / 6 ) );
   v1.push_back( tan ( vc5 ) );
   complex <double> vc6  ( polar ( 1.0,  pi ) );
   v1.push_back( tan ( vc6 ) );

   cout << "The complex components tan (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin() ; Iter1 != v1.end() ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = tan ( c1 ) = (-0.000187346,0.999356)
The modulus of c2 is: 0.999356
The argument of c2 is: 1.57098 radians, which is 90.0107 degrees.

The complex components tan (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(0.713931,0.85004)
(0.24356,0.792403)
(-4.34302e-014,0.761594)
(-0.24356,0.792403)
(-0.713931,0.85004)
(-1.55741,-7.08476e-013)

tanh

Devuelve la tangente hiperbólica de un número complejo.

template <class Type>
complex<Type> tanh(const complex<Type>& complexNum);

Parámetros

complexNum
El número complejo cuya tangente hiperbólica se está determinando.

Valor devuelto

El número complejo que es la tangente hiperbólica del número complejo de entrada.

Comentarios

Identidades que definen la cotangente hiperbólica compleja:

tanh (z) = sinh (z) / cosh (z) = ( exp (z) - exp (- z) ) / ( exp (z) + exp (- z) )

Ejemplo

// complex_tanh.cpp
// compile with: /EHsc
#include <vector>
#include <complex>
#include <iostream>

int main( )
{
   using namespace std;
   double pi = 3.14159265359;
   complex <double> c1 ( 3.0 , 4.0 );
   cout << "Complex number c1 = " << c1 << endl;

   // Values of cosine of a complex number c1
   complex <double> c2 = tanh ( c1 );
   cout << "Complex number c2 = tanh ( c1 ) = " << c2 << endl;
   double absc2 = abs ( c2 );
   double argc2 = arg ( c2 );
   cout << "The modulus of c2 is: " << absc2 << endl;
   cout << "The argument of c2 is: "<< argc2 << " radians, which is "
        << argc2 * 180 / pi << " degrees." << endl << endl;

   // Hyperbolic tangents of the standard angles
   // in the first two quadrants of the complex plane
   vector <complex <double> > v1;
   vector <complex <double> >::iterator Iter1;
   complex <double> vc1  ( polar ( 1.0, pi / 6 ) );
   v1.push_back( tanh ( vc1 ) );
   complex <double> vc2  ( polar ( 1.0, pi / 3 ) );
   v1.push_back( tanh ( vc2 ) );
   complex <double> vc3  ( polar ( 1.0, pi / 2 ) );
   v1.push_back( tanh ( vc3 ) );
   complex <double> vc4  ( polar ( 1.0, 2 * pi / 3 ) );
   v1.push_back( tanh ( vc4 ) );
   complex <double> vc5  ( polar ( 1.0, 5 * pi / 6 ) );
   v1.push_back( tanh ( vc5 ) );
   complex <double> vc6  ( polar ( 1.0, pi ) );
   v1.push_back( tanh ( vc6 ) );

   cout << "The complex components tanh (vci), where abs (vci) = 1"
        << "\n& arg (vci) = i * pi / 6 of the vector v1 are:\n" ;
   for ( Iter1 = v1.begin( ) ; Iter1 != v1.end( ) ; Iter1++ )
      cout << *Iter1 << endl;
}
Complex number c1 = (3,4)
Complex number c2 = tanh ( c1 ) = (1.00071,0.00490826)
The modulus of c2 is: 1.00072
The argument of c2 is: 0.00490474 radians, which is 0.281021 degrees.

The complex components tanh (vci), where abs (vci) = 1
& arg (vci) = i * pi / 6 of the vector v1 are:
(0.792403,0.24356)
(0.85004,0.713931)
(-3.54238e-013,1.55741)
(-0.85004,0.713931)
(-0.792403,0.24356)
(-0.761594,-8.68604e-014)