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C Programming/complex.h

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complex.h is a header file that defines functionality for complex arithmetic.[1] These functions use the built-in type complex which was introduced with the C99 revision of C.

This header should not be confused with the C++ standard library header <complex>, which implements complex numbers in a completely different way (as a template class, complex<T>).


Universal macro:

  • complex - alias of _Complex
  • _Complex_I - constant, which has type "const float _Complex" and value of imaginary unit I, I*I=-1

Defined only for `C99 Annex G`-compatible compiler modes - native support of imaginary types (it is very rare because Annex G is mathematically incorrect[2]):

  • imaginary - alias of _Imaginary
  • _Imaginary_I Expands to a constant expression of type const float _Imaginary with the value of the imaginary unit.

Universal alias of I:

  • I - alias of _Imaginary_I if it is defined else - alias of _Complex_I

Application may undefine complex, imaginary, and I macros if they interfere with internal values.


Each function declared in complex.h has three versions, each of which works with a different floating-point type (double, float and long double). Only the double version of each function is listed here; to use the float (or long double) version, append an f (or an l, respectively) to the function's name (e.g., cabsf and cabsl).

Note that all angles are in radians.

Basic operations
cabs computes absolute value
carg computes argument of a complex number
cimag computes imaginary part of a complex number
creal computes real part of a complex number
conj computes complex conjugate
cproj computes complex projection into the Riemann sphere
Exponentiation operations
cexp computes complex exponential
clog computes complex logarithm
csqrt computes complex square root
cpow computes complex power
Trigonometric operations
csin computes complex sine
ccos computes complex cosine
ctan computes complex tangent
casin computes complex arc sine
cacos computes complex arc cosine
catan computes complex arc tangent
Hyperbolic operations
csinh computes complex hyperbolic sine
ccosh computes complex hyperbolic cosine
ctanh computes complex hyperbolic tangent
casinh computes complex hyperbolic arc sine
cacosh computes complex hyperbolic arc cosine
catanh computes complex hyperbolic arc tangent


  1. complex.h: complex arithmetic – Base Definitions Reference, The Single UNIX® Specification, Issue 7 from The Open Group
  2. Dr. Dobb's journal: software tools for the professional programmer, Volume 27 "many results in Annex G are pure invention and mathematically incorrect"