`<complex>` 関数

[アーティクル]
10/12/2023

abs

複素数の係数を計算します。

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

パラメーター

complexNum
係数を特定する複素数。

戻り値

複素数の係数。

解説

複素数の係数は、複素数を表すベクトル長の測定単位です。複素数 a + bi の剰余は(a2^{b² + )の平方根}であり、|a + bi|. 複素数 a + bi のノルムは (a^{2 +}b^{2) である。} 複素数のノルムは、その係数の平方根となります。

例

// 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

複素数から偏角を抽出します。

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

パラメーター

complexNum
偏角を特定する複素数。

戻り値

複素数の偏角。

解説

偏角は、複素平面における正の実軸と複素ベクトルのなす角度です。複素数 a + bi の場合、偏角は arctan( b/a) と等しくなります。角度は、正の実軸から反時計方向に測定した場合は正方向、時計方向に測定した場合は負方向になります。主値は -pi より大きく、+pi 以下となります。

例

// 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

複素数の複素共役を返します。

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

パラメーター

complexNum
複素共役が返される複素数。

戻り値

入力された複素数の複素共役。

解説

複素数 a + bi の複素共役は a - bi です。複素数とその共役の積は、数値 a2 + b2 のノルムです。

例

// 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

複素数のコサインを返します。

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

パラメーター

complexNum
コサインを特定する複素数。

戻り値

入力された複素数のコサインである複素数。

解説

複素数のコサインを定義する単位元は次のようになります。

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

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

例

// 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

複素数のハイパーボリックコサインを返します。

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

パラメーター

complexNum
ハイパーボリックコサインを特定する複素数。

戻り値

入力された複素数のハイパーボリックコサインである複素数。

解説

複素数のハイパーボリックコサインを定義する単位元は次のようになります。

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

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

例

// 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

複素数の指数関数を返します。

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

パラメーター

complexNum
指数を特定する複素数。

戻り値

入力された複素数の指数である複素数。

例

// 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

複素数の虚数部を抽出します。

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

パラメーター

complexNum
実数部が抽出される複素数。

戻り値

グローバル関数としての複素数の虚数部。

解説

このテンプレート関数は、複素数の実数部を変更するのに使用することはできません。実数部を変更するには、新しい複素数に成分値を割り当てる必要があります。

例

// 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

複素数の自然対数を返します。

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

パラメーター

complexNum
自然対数を特定する複素数。

戻り値

入力された複素数の自然対数である複素数。

解説

分岐切断線は負の実軸に沿っています。

例

// 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

複素数の底 10 の対数を返します。

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

パラメーター

complexNum
底 10 の対数を特定する複素数。

戻り値

入力された複素数の底 10 の対数である複素数。

解説

分岐切断線は負の実軸に沿っています。

例

// 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

複素数のノルムを抽出します。

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

パラメーター

complexNum
ノルムを特定する複素数。

戻り値

複素数のノルム。

解説

複素数 a + bi のノルムは (a^{2 +}b^{2) である。} 複素数のノルムは、その係数の平方根となります。複素数の係数は、複素数を表すベクトル長の測定単位です。複素数 a + bi の剰余は(a2^{b² + )の平方根}であり、|a + bi|.

例

// 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

デカルト形式で、指定した係数と偏角に対応する複素数を返します。

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

パラメーター

_Modulus
入力される複素数の係数。

_Argument
入力される複素数の偏角。

戻り値

極形式で指定されたデカルト形式の複素数。

解説

極形式の複素数では、係数 r と引数 p を提供します。これらのパラメーターは、a = r * cos p と b = r * sin p の方程式で算出される実数と虚数のデカルト成分 a と b に関連します。

例

// 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

複素数の底を、別の複素数で累乗することによって得られる複素数を評価します。

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);

パラメーター

_Base
複素数、または底をメンバー関数で累乗する複素数のパラメーター型の数値。

_Power
整数または複素数、あるいは底をメンバー関数で累乗する場合の指数である複素数のパラメーター型の数値。

戻り値

指定された底を指定された指数で累乗して得られた複素数。

解説

関数はそれぞれ効率的に両方のオペランドを戻り値の型に変換してから、変換した left の right 乗を返します。

分岐切断線は負の実軸に沿っています。

例

// 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

複素数の実数部を抽出します。

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

パラメーター

complexNum
実数部が抽出される複素数。

戻り値

グローバル関数としての複素数の実数部。

解説

例

// 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

複素数のサインを返します。

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

パラメーター

complexNum
サインを特定する複素数。

戻り値

入力された複素数のサインである複素数。

解説

複素数のサインを定義する単位元は次のようになります。

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

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

例

// 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

複素数のハイパーボリックサインを返します。

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

パラメーター

complexNum
ハイパーボリックサインを特定する複素数。

戻り値

入力された複素数のハイパーボリックサインである複素数。

解説

複素数のハイパーボリックサインを定義する単位元は次のようになります。

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

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

例

// 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

複素数の平方根を計算します。

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

パラメーター

complexNum
平方根を求める対象の複素数。

戻り値

複素数の平方根。

解説

平方根の半開区間には位相角があります (-pi/2, pi/2)。

複素平面の分岐切断線は負の実軸に沿っています。

複素数の平方根には係数 (入力数の平方根) と偏角 (入力数の半分) があります。

例

// 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

複素数のタンジェントを返します。

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

パラメーター

complexNum
タンジェントを特定する複素数。

戻り値

入力された複素数のタンジェントである複素数。

解説

複素数のコタンジェントを定義する単位元は次のようになります。

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

例

// 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

複素数のハイパーボリックタンジェントを返します。

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

パラメーター

complexNum
ハイパーボリックタンジェントを特定する複素数。

戻り値

入力された複素数のハイパーボリックタンジェントである複素数。

解説

複素数のハイパーボリックコタンジェントを定義する単位元は次のようになります。

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

例

// 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)

<complex> 関数

abs

パラメーター

戻り値

解説

例

acos

acosh

arg

パラメーター

戻り値

解説

例

asin

asinh

atan

atanh

conj

パラメーター

戻り値

解説

例

cos

パラメーター

戻り値

解説

例

cosh

パラメーター

戻り値

解説

例

exp

パラメーター

戻り値

例

imag

パラメーター

戻り値

解説

例

log

パラメーター

戻り値

解説

例

log10

パラメーター

戻り値

解説

例

norm

パラメーター

戻り値

解説

例

polar

パラメーター

戻り値

解説

例

pow

パラメーター

戻り値

解説

例

proj

real

パラメーター

戻り値

解説

例

sin

パラメーター

戻り値

解説

例

sinh

パラメーター

戻り値

`<complex>` 関数