Funções <complex>
abs
Calcula o módulo de um número complexo.
template <class Type>
Type abs(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo módulo deve ser determinado.
Valor de retorno
O módulo de um número complexo.
Comentários
O módulo de um número complexo é uma medida do comprimento do vetor que representa o número complexo. O módulo de um número complexo a + bi é a raiz quadrada de (a2 + b2), escrito | a + bi|. A norma de um número complexo a + bi é (a2 + b2). A norma de um número complexo é o quadrado do seu módulo.
Exemplo
// 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
Extrai o argumento de um número complexo.
template <class Type>
Type arg(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo argumento deve ser determinado.
Valor de retorno
O argumento do número complexo.
Comentários
O argumento é o ângulo que o vetor complexo faz com o eixo real positivo no plano complexo. Para um número complexo, a + bi, o argumento é igual a arctan(b/a). O ângulo tem valor positivo quando medido no sentido anti-horário do eixo real positivo e valor negativo quando medido no sentido horário. Os valores de entidade de segurança são maiores que -pi e menores ou iguais a +pi.
Exemplo
// 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
Retorna o conjugado complexo de um número complexo.
template <class Type>
complex<Type> conj(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo conjugado complexo está sendo retornado.
Valor de retorno
O conjugado complexo do número complexo de entrada.
Comentários
O conjugado complexo de um número complexo, a + bi, é a – bi. O produto de um número complexo e seu conjugado é a norma do número a2 + b2.
Exemplo
// 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
Retorna o cosseno de um número complexo.
template <class Type>
complex<Type> cos(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo cosseno está sendo determinado.
Valor de retorno
O número complexo que é o cosseno do número complexo de entrada.
Comentários
Identidades que definem os cossenos complexos:
cos (z) = (1/2)*(exp (iz) + exp (- iz) )
cos (z) = cos (a + bi) = cos (a) cosh ( b) - isin (a) sinh (b)
Exemplo
// 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
Retorna o cosseno hiperbólico de um número complexo.
template <class Type>
complex<Type> cosh(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo cosseno hiperbólico está sendo determinado.
Valor de retorno
O número complexo que é o cosseno hiperbólico do número complexo de entrada.
Comentários
Identidades que definem os cossenos hiperbólicos complexos:
cos (z) = (1/2)*( exp (z) + exp (- z) )
cos (z) = cosh (a + bi) = cosh (a) cos (b) + isinh (a) sin (b)
Exemplo
// 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
Retorna a função exponencial de um número complexo.
template <class Type>
complex<Type> exp(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo exponencial está sendo determinado.
Valor de retorno
O número complexo que é o exponencial do número complexo de entrada.
Exemplo
// 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
Extrai o componente imaginário de um número complexo.
template <class Type>
Type imag(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja parte real deve ser extraída.
Valor de retorno
A parte imaginária do número complexo como uma função global.
Comentários
Essa função de modelo não pode ser usada para modificar a parte real do número complexo. Para alterar a parte real, o valor do componente deve ser atribuído a um novo número complexo.
Exemplo
// 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
Retorna o logaritmo natural de um número complexo.
template <class Type>
complex<Type> log(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo logaritmo natural está sendo determinado.
Valor de retorno
O número complexo que é o logaritmo natural do número complexo de entrada.
Comentários
Os cortes de branch são feitos ao longo do eixo real negativo.
Exemplo
// 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
Retorna o logaritmo de base 10 de um número complexo.
template <class Type>
complex<Type> log10(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo logaritmo de base 10 está sendo determinado.
Valor de retorno
O número complexo que é o logaritmo de base 10 do número complexo de entrada.
Comentários
Os cortes de branch são feitos ao longo do eixo real negativo.
Exemplo
// 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
Extrai a norma de um número complexo.
template <class Type>
Type norm(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja norma deve ser determinada.
Valor de retorno
A norma de um número complexo.
Comentários
A norma de um número complexo a + bi é (a2 + b2). A norma de um número complexo é o quadrado do seu módulo. O módulo de um número complexo é uma medida do comprimento do vetor que representa o número complexo. O módulo de um número complexo a + bi é a raiz quadrada de (a2 + b2), escrito | a + bi|.
Exemplo
// 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
Retorna o número complexo, que corresponde a um módulo e um argumento especificado, na forma cartesiana.
template <class Type>
complex<Type> polar(const Type& _Modulus, const Type& _Argument = 0);
Parâmetros
_Modulus
O módulo do número complexo que está sendo inserido.
_Argument
O argumento do número complexo que está sendo inserido.
Valor de retorno
Formulário cartesiano do número complexo especificado na forma polar.
Comentários
A forma polar de um número complexo fornece o módulo r e o argumento p, em que esses parâmetros são relacionados aos componentes cartesianos reais e imaginários a e b pelas equações a = r * cos p e b = r * sin p.
Exemplo
// 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
Avalia o número complexo obtido elevando uma base que é um número complexo à potência de outro número complexo.
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);
Parâmetros
_Base
O número complexo ou número do tipo de parâmetro do número complexo que é a base a ser elevada a uma potência pela função membro.
_Power
O inteiro ou número complexo ou número do tipo de parâmetro do número complexo que é a potência à qual a base deve ser elevada pela função membro.
Valor de retorno
O número complexo obtido elevando a base especificada à potência especificada.
Comentários
As funções convertem efetivamente ambos os operandos para o tipo retornado e então retornam a esquerda convertida à potência direita.
O corte de branch está junto com o eixo real negativo.
Exemplo
// 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
Extrai o componente real de um número complexo.
template <class Type>
Type real(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja parte real deve ser extraída.
Valor de retorno
A parte real do número complexo como uma função global.
Comentários
Essa função de modelo não pode ser usada para modificar a parte real do número complexo. Para alterar a parte real, o valor do componente deve ser atribuído a um novo número complexo.
Exemplo
// 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
Retorna o seno de um número complexo.
template <class Type>
complex<Type> sin(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo seno está sendo determinado.
Valor de retorno
O número complexo que é o seno do número complexo de entrada.
Comentários
Identidades que definem os senos complexos:
sin (z) = (1/2 i)*( exp (iz) - exp (- iz) )
sin (z) = sin (a + bi) = sin (a) cosh (b) + icos (a) sinh (b)
Exemplo
// 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
Retorna o seno hiperbólico de um número complexo.
template <class Type>
complex<Type> sinh(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cujo seno hiperbólico está sendo determinado.
Valor de retorno
O número complexo que é o seno hiperbólico do número complexo de entrada.
Comentários
Identidades que definem os senos hiperbólicos complexos:
sinh (z) = (1/2)*( exp (z) - exp (- z) )
sinh (z) = sinh (a + bi) = sinh (a) cos (b) + icosh (a) sin (b)
Exemplo
// 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 a raiz quadrada de um número complexo.
template <class Type>
complex<Type> sqrt(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja raiz quadrada deve ser encontrada.
Valor de retorno
A raiz quadrada de um número complexo.
Comentários
A raiz quadrada terá um ângulo de fase no intervalo semiaberto (-pi/2, pi/2].
Os cortes de branch no plano complexo são feitos ao longo do eixo real negativo.
A raiz quadrada de um número complexo terá um módulo que é a raiz quadrada do número de entrada e um argumento que é uma metade desse número de entrada.
Exemplo
// 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
Retorna a tangente de um número complexo.
template <class Type>
complex<Type> tan(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja tangente está sendo determinada.
Valor de retorno
O número complexo que é a tangente do número complexo de entrada.
Comentários
Identidades que definem a cotangente complexa:
tan (z) = sin (z) / cos (z) = ( exp (iz) - exp (- iz) ) / i( exp (iz) + exp (- iz) )
Exemplo
// 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
Retorna a tangente hiperbólica de um número complexo.
template <class Type>
complex<Type> tanh(const complex<Type>& complexNum);
Parâmetros
complexNum
O número complexo cuja tangente hiperbólica está sendo determinada.
Valor de retorno
O número complexo que é a tangente hiperbólica do número complexo de entrada.
Comentários
Identidades que definem a cotangente hiperbólica complexa:
tanh (z) = sinh (z) / cosh (z) = ( exp (z) - exp (- z) ) / ( exp (z) + exp (- z) )
Exemplo
// 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)