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Difference between revisions of "cpp/numeric/complex/tan"

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< cpp‎ | numeric‎ | complex
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m (→‎Return value: +{box})
 
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{{cpp/title|tan{{small|(std::complex)}}}}
 
{{cpp/title|tan{{small|(std::complex)}}}}
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{{cpp/numeric/complex/sidebar}}
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{{cpp/numeric/complex/navbar}}
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{{ddcl | header=complex | 1=
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{{ddcl|header=complex|1=
 
template< class T >  
 
template< class T >  
 
complex<T> tan( const complex<T>& z );
 
complex<T> tan( const complex<T>& z );
 
}}
 
}}
  
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Computes complex tangent of a complex value {{tt|z}}.
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Computes complex tangent of a complex value {{c|z}}.
  
 
===Parameters===
 
===Parameters===
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{{param list begin}}
+
{{par begin}}
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{{param list item | z | complex value}}
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{{par|z|complex value}}
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{{param list end}}
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{{par end}}
  
 
===Return value===
 
===Return value===
 +
If no errors occur, the complex tangent of {{c|z}} is returned.
  
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Complex tangent of {{tt|z}}.
+
Errors and special cases are handled as if the operation is implemented by {{box|{{c/core|-i * }}{{nbspt}}{{ltt std|cpp/numeric/complex/tanh}}{{c/core|(i * z)}}}}, where {{tt|i}} is the imaginary unit.
 +
 
 +
===Notes===
 +
Tangent is an analytical function on the complex plain and has no branch cuts. It is periodic with respect to the real component, with period &pi;i, and has poles of the first order along the real line, at coordinates {{math|(π(1/2 + n), 0)}}. However no common floating-point representation is able to represent π/2 exactly, thus there is no value of the argument for which a pole error occurs.
 +
 
 +
Mathematical definition of the tangent is {{math|tan z {{=}} {{mfrac|i(e{{su|p=-iz}}-e{{su|p=iz}})|e{{su|p=-iz}}+e{{su|p=iz}}}}}}.
 +
 
 +
===Example===
 +
{{example|
 +
|code=
 +
#include <cmath>
 +
#include <complex>
 +
#include <iostream>
 +
 
 +
int main()
 +
{
 +
    std::cout << std::fixed;
 +
    std::complex<double> z(1.0, 0.0); // behaves like real tangent along the real line
 +
    std::cout << "tan" << z << " = " << std::tan(z)
 +
              << " ( tan(1) = " << std::tan(1) << ")\n";
 +
 
 +
    std::complex<double> z2(0.0, 1.0); // behaves like tanh along the imaginary line
 +
    std::cout << "tan" << z2 << " = " << std::tan(z2)
 +
              << " (tanh(1) = " << std::tanh(1) << ")\n";
 +
}
 +
|output=
 +
tan(1.000000,0.000000) = (1.557408,0.000000) ( tan(1) = 1.557408)
 +
tan(0.000000,1.000000) = (0.000000,0.761594) (tanh(1) = 0.761594)
 +
}}
  
 
===See also===
 
===See also===
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{{dcl list begin}}
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{{dsc begin}}
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{{dcl list template | cpp/numeric/complex/dcl list sin}}
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{{dsc inc|cpp/numeric/complex/dsc sin}}
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{{dcl list template | cpp/numeric/complex/dcl list cos}}
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{{dsc inc|cpp/numeric/complex/dsc cos}}
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{{dcl list end}}
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{{dsc inc|cpp/numeric/complex/dsc atan}}
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{{dsc inc|cpp/numeric/math/dsc tan}}
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{{dsc inc|cpp/numeric/valarray/dsc tan}}
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{{dsc see c|c/numeric/complex/ctan}}
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{{dsc end}}
 +
 
 +
{{langlinks|de|es|fr|it|ja|pt|ru|zh}}

Latest revision as of 10:58, 22 April 2023

 
C++
 
Numerics library
Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array (valarray)
Pseudo-random number generation
Factor operations
(C++17)
(C++17)
Interpolations
(C++20)
(C++20)
Saturation arithmetic
(C++26)
(C++26)
(C++26)
(C++26)
(C++26)

Generic numeric operations
Bit operations
(C++20)    
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++23)
(C++20)
 
std::complex
 
Defined in header <complex>
template< class T >
complex<T> tan( const complex<T>& z );

Computes complex tangent of a complex value z.

Contents

[edit] Parameters

z - complex value

[edit] Return value

If no errors occur, the complex tangent of z is returned.

Errors and special cases are handled as if the operation is implemented by -i * std::tanh(i * z), where i is the imaginary unit.

[edit] Notes

Tangent is an analytical function on the complex plain and has no branch cuts. It is periodic with respect to the real component, with period πi, and has poles of the first order along the real line, at coordinates (π(1/2 + n), 0). However no common floating-point representation is able to represent π/2 exactly, thus there is no value of the argument for which a pole error occurs.

Mathematical definition of the tangent is tan z =
i(e-iz-eiz)
e-iz+eiz
.

[edit] Example

Run this code
#include <cmath>
#include <complex>
#include <iostream>
 
int main()
{
    std::cout << std::fixed;
    std::complex<double> z(1.0, 0.0); // behaves like real tangent along the real line
    std::cout << "tan" << z << " = " << std::tan(z)
              << " ( tan(1) = " << std::tan(1) << ")\n";
 
    std::complex<double> z2(0.0, 1.0); // behaves like tanh along the imaginary line
    std::cout << "tan" << z2 << " = " << std::tan(z2)
              << " (tanh(1) = " << std::tanh(1) << ")\n";
}

Output: 

tan(1.000000,0.000000) = (1.557408,0.000000) ( tan(1) = 1.557408)
tan(0.000000,1.000000) = (0.000000,0.761594) (tanh(1) = 0.761594)

[edit] See also

 computes sine of a complex number (sin(z))
(function template) [edit]
computes cosine of a complex number (cos(z))
(function template) [edit]
(C++11)
 computes arc tangent of a complex number (arctan(z))
(function template) [edit]
(C++11)(C++11)
 computes tangent (tan(x))
(function) [edit]
applies the function std::tan to each element of valarray
(function template) [edit]
C documentation for ctan
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