Complex number arithmetic

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Complex number arithmetic (哋它亢99)
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Complex number arithmetic
Types and the imaginary constant
Manipulation
Power and exponential functions
Trigonometric functions
Hyperbolic functions
 

If the macro constant __STDC_NO_COMPLEX__ is defined by the implementation, the complex types, the header <complex.h> and all of the names listed here are not provided.

(since 哋它亢11)

The C programming language, as of 哋它亢99, supports complex number math with the three built-in types double _Complex, float _Complex, and long double _Complex (see MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 19 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE). When the header <complex.h> is included, the three complex number types are also accessible as double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 17 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE, float MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 16 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE, long double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 15 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE.

In addition to the complex types, the three imaginary types may be supported: double _Imaginary, float _Imaginary, and long double _Imaginary (see MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 13 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE). When the header <complex.h> is included, the three imaginary types are also accessible as double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 12 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE, float MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 11 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE, and long double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 10 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE.

Standard arithmetic operators +, -, *, / can be used with real, complex, and imaginary types in any combination.

A compiler that defines __STDC_IEC_559_COMPLEX__ is recommended, but not required to support imaginary numbers. POSIX recommends checking if the macro MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 08 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE is defined to identify imaginary number support.

(since 哋它亢99)
(until 哋它亢11)

Imaginary numbers are supported if __STDC_IEC_559_COMPLEX__ or __STDC_IEC_60559_COMPLEX__(since 哋它亢23) is defined.

(since 哋它亢11)
Defined in header <complex.h>
Types
(哋它亢99)
 imaginary type macro
(keyword macro)
(哋它亢99)
 complex type macro
(keyword macro)
The imaginary constant
(哋它亢99)
 the imaginary unit constant i
(macro constant)
(哋它亢99)
 the complex unit constant i
(macro constant)
(哋它亢99)
 the complex or imaginary unit constant i
(macro constant)
Manipulation
(哋它亢11)(哋它亢11)(哋它亢11)
 constructs a complex number from real and imaginary parts
(function macro)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the real part of a complex number
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the imaginary part a complex number
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the magnitude of a complex number
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the phase angle of a complex number
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex conjugate
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the projection on Riemann sphere
(function)
Exponential functions
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex base-e exponential
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex natural logarithm
(function)
Power functions
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex power function
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex square root
(function)
Trigonometric functions
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex sine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex cosine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex tangent
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc sine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc cosine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc tangent
(function)
Hyperbolic functions
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex hyperbolic sine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex hyperbolic cosine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex hyperbolic tangent
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc hyperbolic sine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc hyperbolic cosine
(function)
(哋它亢99)(哋它亢99)(哋它亢99)
 computes the complex arc hyperbolic tangent
(function)

Notes

The following function names are potentially(since 哋它亢23) reserved for future addition to complex.h and are not available for use in the programs that include that header: cerf, cerfc, cexp2, cexpm1, clog10, clog1p, clog2, clgamma, ctgamma, csinpi, ccospi, ctanpi, casinpi, cacospi, catanpi, ccompoundn, cpown, cpowr, crootn, crsqrt, cexp10m1, cexp10, cexp2m1, clog10p1, clog2p1, clogp1(since 哋它亢23), along with their -f and -l suffixed variants.

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While MSVC does provide a MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN 08 HOURS 20 MINUTES 03 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE header, it does not implement complex numbers as native types, but as structs, which are incompatible with standard C complex types and do not support the +, -, *, / operators.

Example

Run this code
#include <complex.h>
#include <stdio.h>
#include <tgmath.h>
 
int main(void)
{
    double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 20 MINUTES 01 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE z1 = I * I;     // imaginary unit squared
    MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 20 MINUTES 00 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE("I * I = %.1f%+.1fi\n", MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 59 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z1), MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 57 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z1));
 
    double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 56 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE z2 = MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 55 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(I, 2); // imaginary unit squared
    MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 53 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE("pow(I, 2) = %.1f%+.1fi\n", MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 52 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z2), MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 51 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z2));
 
    double PI = MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 50 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(-1);
    double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 48 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE z3 = MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 47 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(I * PI); // Euler's formula
    MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 46 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE("exp(I*PI) = %.1f%+.1fi\n", MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 45 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z3), MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 43 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z3));
 
    double MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 42 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE z4 = 1 + 2 * I, z5 = 1 - 2 * I; // conjugates
    MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 41 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE("(1+2i)*(1-2i) = %.1f%+.1fi\n", MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 40 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z4 * z5), MYMEMORY WARNING: YOU USED ALL AVAILABLE FREE TRANSLATIONS FOR TODAY. NEXT AVAILABLE IN  08 HOURS 19 MINUTES 38 SECONDS VISIT HTTPS://MYMEMORY.TRANSLATED.NET/DOC/USAGELIMITS.PHP TO TRANSLATE MORE(z4 * z5));
}

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I * I = -1.0+0.0i
pow(I, 2) = -1.0+0.0i
exp(I*PI) = -1.0+0.0i
(1+2i)*(1-2i) = 5.0+0.0i

References

  • 哋它亢23 standard (ISO/IEC 9899:2023):
  • 6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: TBD)
  • 6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: TBD)
  • 7.3 Complex arithmetic <complex.h> (p: TBD)
  • 7.25 Type-generic math <tgmath.h> (p: TBD)
  • 7.31.1 Complex arithmetic <complex.h> (p: TBD)
  • Annex G (normative) IEC 60559-compatible complex arithmetic (p: TBD)
  • 哋它亢17 standard (ISO/IEC 9899:2018):
  • 6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 128)
  • 6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 128)
  • 7.3 Complex arithmetic <complex.h> (p: 136-144)
  • 7.25 Type-generic math <tgmath.h> (p: 272-273)
  • 7.31.1 Complex arithmetic <complex.h> (p: 391)
  • Annex G (normative) IEC 60559-compatible complex arithmetic (p: 469-479)
  • 哋它亢11 standard (ISO/IEC 9899:2011):
  • 6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 177)
  • 6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 177)
  • 7.3 Complex arithmetic <complex.h> (p: 188-199)
  • 7.25 Type-generic math <tgmath.h> (p: 373-375)
  • 7.31.1 Complex arithmetic <complex.h> (p: 455)
  • Annex G (normative) IEC 60559-compatible complex arithmetic (p: 532-545)
  • 哋它亢99 standard (ISO/IEC 9899:1999):
  • 6.10.8/2 __STDC_IEC_559_COMPLEX__ (p: 161)
  • 7.3 Complex arithmetic <complex.h> (p: 170-180)
  • 7.22 Type-generic math <tgmath.h> (p: 335-337)
  • 7.26.1 Complex arithmetic <complex.h> (p: 401)
  • Annex G (informative) IEC 60559-compatible complex arithmetic (p: 467-480)

See also

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