fonctions<complex>
abs
Calcule le module d'un nombre complexe.
template <class Type>
Type abs(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le module doit être déterminé.
Valeur de retour
Module d’un nombre complexe.
Notes
Le module d’un nombre complexe est une mesure de la longueur du vecteur qui représente le nombre complexe. Le modulus d’un nombre complexe a + bi est la racine carrée de (a2 + b2), écrit |a + bi|. La norme d’un nombre complexe a + bi est (a2 + b2). La norme d’un nombre complexe est le carré de son module.
Exemple
// 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>&);
arg
Extrait l’argument d’un nombre complexe.
template <class Type>
Type arg(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont l’argument doit être déterminé.
Valeur de retour
Argument du nombre complexe.
Notes
L’argument est l’angle que le vecteur complexe effectue avec l’axe réel positif dans le plan complexe. Pour un nombre complexe a + bi, l’argument est égal à arctan(b/a). L’angle a un sens positif quand il est mesuré dans le sens contraire des aiguilles d’une montre à partir de l’axe réel positif et un sens négatif quand il est mesuré dans le sens des aiguilles d’une montre. Les valeurs principales sont supérieures à -pi et inférieures ou égales à +pi.
Exemple
// 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
Retourne le conjugué complexe d'un nombre complexe.
template <class Type>
complex<Type> conj(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le conjugué complexe est retourné.
Valeur de retour
Conjugué complexe du nombre complexe d’entrée.
Notes
Le conjugue complexe d’un nombre complexe a + bi est un - bi. Le produit d’un nombre complexe et son conjugué est la norme du nombre a2 + b2.
Exemple
// 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
Retourne le cosinus d'un nombre complexe.
template <class Type>
complex<Type> cos(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le cosinus est déterminé.
Valeur de retour
Nombre complexe qui correspond au cosinus du nombre complexe d’entrée.
Notes
Identités définissant les cosinus complexes :
cos (z) = (1/2)*(exp (iz) + exp (- iz) )
cos (z) = cos (a + bi) = cos (a) cosh ( b) - isin (a) sinh (b)
Exemple
// 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
Retourne le cosinus hyperbolique d'un nombre complexe.
template <class Type>
complex<Type> cosh(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le cosinus hyperbolique est déterminé.
Valeur de retour
Nombre complexe qui correspond au cosinus hyperbolique du nombre complexe d’entrée.
Notes
Identités définissant les cosinus hyperboliques complexes :
cos (z) = (1/2)*( exp (z) + exp (- z) )
cos (z) = cosh (a + bi) = cosh (a) cos (b) + isinh (a) sin (b)
Exemple
// 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
Retourne la fonction exponentielle d'un nombre complexe.
template <class Type>
complex<Type> exp(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la valeur exponentielle est déterminée.
Valeur de retour
Nombre complexe qui correspond à la valeur exponentielle du nombre complexe d’entrée.
Exemple
// 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
Extrait le composant imaginaire d'un nombre complexe.
template <class Type>
Type imag(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la partie réelle doit être extraite.
Valeur de retour
Partie imaginaire du nombre complexe en tant que fonction globale.
Notes
Cette fonction de modèle ne peut pas être utilisée pour modifier la partie réelle du nombre complexe. Pour modifier la partie réelle, un nouveau nombre complexe doit être affecté à la valeur du composant.
Exemple
// 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
Retourne le logarithme naturel d'un nombre complexe.
template <class Type>
complex<Type> log(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le logarithme naturel est déterminé.
Valeur de retour
Nombre complexe qui correspond au logarithme naturel du nombre complexe d’entrée.
Notes
Les coupures de branche suivent l’axe réel négatif.
Exemple
// 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
Retourne le logarithme de base 10 d'un nombre complexe.
template <class Type>
complex<Type> log10(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le logarithme base 10 est déterminé.
Valeur de retour
Nombre complexe qui correspond au logarithme base 10 du nombre complexe d’entrée.
Notes
Les coupures de branche suivent l’axe réel négatif.
Exemple
// 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
Extrait la norme d'un nombre complexe.
template <class Type>
Type norm(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la norme doit être déterminée.
Valeur de retour
Norme d’un nombre complexe.
Notes
La norme d’un nombre complexe a + bi est (a2 + b2). La norme d’un nombre complexe est le carré de son module. Le module d’un nombre complexe est une mesure de la longueur du vecteur qui représente le nombre complexe. Le modulus d’un nombre complexe a + bi est la racine carrée de (a2 + b2), écrit |a + bi|.
Exemple
// 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
Retourne le nombre complexe qui correspond à un module et à un argument spécifiés, au format cartésien.
template <class Type>
complex<Type> polar(const Type& _Modulus, const Type& _Argument = 0);
Paramètres
_Modulus
Module du nombre complexe entré.
_Argument
Argument du nombre complexe entré.
Valeur de retour
Forme cartésienne du nombre complexe spécifiée en forme polaire.
Notes
La forme polaire d’un nombre complexe fournit le module r et l’argument p, où ces paramètres sont liés aux composants cartésiens réels et imaginaires a et b par les équations a = r * cos p et b = r * sin p.
Exemple
// 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
Évalue le nombre complexe obtenu en élevant une base qui est un nombre complexe à la puissance d'un autre nombre complexe.
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);
Paramètres
_Base
Nombre complexe ou nombre qui a le même type de paramètre que le nombre complexe, lequel est la base à élever à une puissance par la fonction membre.
_Power
Entier ou nombre complexe ou nombre qui a le même type de paramètre que le nombre complexe, lequel est la puissance à laquelle la base doit être élevée par la fonction membre.
Valeur de retour
Nombre complexe obtenu en élevant la base spécifiée à la puissance spécifiée.
Notes
Les fonctions convertissent chacune les deux opérandes dans le type de retour, puis retournent l’opérande de gauche converti à la puissance de l’opérande de droite.
La coupure de branche suit l’axe réel négatif.
Exemple
// 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
Extrait le composant réel d'un nombre complexe.
template <class Type>
Type real(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la partie réelle doit être extraite.
Valeur de retour
Partie réelle du nombre complexe en tant que fonction globale.
Notes
Cette fonction de modèle ne peut pas être utilisée pour modifier la partie réelle du nombre complexe. Pour modifier la partie réelle, un nouveau nombre complexe doit être affecté à la valeur du composant.
Exemple
// 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
Retourne le sinus d'un nombre complexe.
template <class Type>
complex<Type> sin(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le sinus est déterminé.
Valeur de retour
Nombre complexe qui correspond au sinus du nombre complexe d’entrée.
Notes
Identités définissant les sinus complexes :
sin (z) = (1/2 i)*( exp (iz) - exp (- iz) )
sin (z) = sin (a + bi) = sin (a) cosh (b) + icos (a) sinh (b)
Exemple
// 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
Retourne le sinus hyperbolique d'un nombre complexe.
template <class Type>
complex<Type> sinh(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont le sinus hyperbolique est déterminé.
Valeur de retour
Nombre complexe qui correspond au sinus hyperbolique du nombre complexe d’entrée.
Notes
Identités définissant les sinus hyperboliques complexes :
sinh (z) = (1/2)*( exp (z) - exp (- z) )
sinh (z) = sinh (a + bi) = sinh (a) cos (b) + i cosh(a) sin (b)
Exemple
// 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
Calcule la racine carrée d’un nombre complexe.
template <class Type>
complex<Type> sqrt(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre dont la racine carrée est recherchée.
Valeur de retour
Racine carrée d’un nombre complexe.
Notes
La racine carrée a un angle de phase dans l’intervalle à moitié ouvert (-pi/2, pi/2].
Les coupures de branche dans le plan complexe suivent l’axe réel négatif.
La racine carrée d’un nombre complexe a un module qui est la racine carrée du nombre d’entrée et un argument qui est la moitié du nombre d’entrée.
Exemple
// 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
Retourne la tangente d'un nombre complexe.
template <class Type>
complex<Type> tan(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la tangente est déterminée.
Valeur de retour
Nombre complexe qui correspond à la tangente du nombre complexe d’entrée.
Notes
Identités définissant la cotangente complexe :
tan (z) = sin (z) / cos (z) = ( exp (iz) - exp (- iz) ) / i( exp (iz) + exp (- iz) )
Exemple
// 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
Retourne la tangente hyperbolique d'un nombre complexe.
template <class Type>
complex<Type> tanh(const complex<Type>& complexNum);
Paramètres
complexNum
Nombre complexe dont la tangente hyperbolique est déterminée.
Valeur de retour
Nombre complexe qui correspond à la tangente hyperbolique du nombre complexe d’entrée.
Notes
Identités définissant la cotangente hyperbolique complexe :
tanh (z) = sinh (z) / cosh (z) = ( exp (z) - exp (- z) ) / ( exp (z) + exp (- z) )
Exemple
// 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)