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# tanh

Returns the hyperbolic tangent of a complex number.

## Syntax

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

## Parameters

_ComplexNum

The complex number whose hyperbolic tangent is being determined.

## Return Value

The complex number that is the hyperbolic tangent of the input complex number.

## Remarks

Identities defining the complex hyperbolic cotangent:

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

## Example

// 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)