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Complex number arithmetic

From cppreference.com
< c‎ | numeric
If the macro constant __STDC_NO_COMPLEX__(C11) is defined by the implementation, the complex types, the header <complex.h> and all of the names listed here are not provided. (since C11)

The C programming language, as of C99, supports complex number math with the three built-in types double _Complex, float _Complex, and long double _Complex (see _Complex). When the header <complex.h> is included, the three complex number types are also accessible as double complex, float complex, long double complex.

In addition to the complex types, the three imaginary types may be supported: double _Imaginary, float _Imaginary, and long double _Imaginary (see _Imaginary). When the header <complex.h> is included, the three imaginary types are also accessible as double imaginary, float imaginary, and long double imaginary.

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 _Imaginary_I is defined to identify imaginary number support.

(since C99)
(until C11)

Imaginary numbers are supported if __STDC_IEC_559_COMPLEX__ is defined.

(since C11)
Defined in header <complex.h>

Contents

Types
imaginary type macro
(macro constant) [edit]
complex type macro
(macro constant) [edit]
The imaginary constant
the imaginary unit constant i
(macro constant) [edit]
the complex unit constant i
(macro constant) [edit]
(C99)
the complex or imaginary unit constant i
(macro constant) [edit]
Manipulation
(C11)(C11)(C11)
constructs a complex number from real and imaginary parts
(function macro) [edit]
(C99)(C99)(C99)
computes the real part of a complex number
(function) [edit]
(C99)(C99)(C99)
computes the imaginary part a complex number
(function) [edit]
(C99)(C99)(C99)
computes the magnitude of a complex number
(function) [edit]
(C99)(C99)(C99)
computes the phase angle of a complex number
(function) [edit]
(C99)(C99)(C99)
computes the complex conjugate
(function) [edit]
(C99)(C99)(C99)
computes the projection on Riemann sphere
(function) [edit]
Exponential functions
(C99)(C99)(C99)
computes the complex base-e exponential
(function) [edit]
(C99)(C99)(C99)
computes the complex natural logarithm
(function) [edit]
Power functions
(C99)(C99)(C99)
computes the complex power function
(function) [edit]
(C99)(C99)(C99)
computes the complex square root
(function) [edit]
Trigonometric functions
(C99)(C99)(C99)
computes the complex sine
(function) [edit]
(C99)(C99)(C99)
computes the complex cosine
(function) [edit]
(C99)(C99)(C99)
computes the complex tangent
(function) [edit]
(C99)(C99)(C99)
computes the complex arc sine
(function) [edit]
(C99)(C99)(C99)
computes the complex arc cosine
(function) [edit]
(C99)(C99)(C99)
computes the complex arc tangent
(function) [edit]
Hyperbolic functions
(C99)(C99)(C99)
computes the complex hyperbolic sine
(function) [edit]
(C99)(C99)(C99)
computes the complex hyperbolic cosine
(function) [edit]
(C99)(C99)(C99)
computes the complex hyperbolic tangent
(function) [edit]
(C99)(C99)(C99)
computes the complex arc hyperbolic sine
(function) [edit]
(C99)(C99)(C99)
computes the complex arc hyperbolic cosine
(function) [edit]
(C99)(C99)(C99)
computes the complex arc hyperbolic tangent
(function) [edit]

[edit] Notes

The following function names are 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, and ctgamma, along with their -f and -l suffixed variants.

Although the C standard names the inverse hyperbolics with "complex arc hyperbolic sine" etc., the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct names are "complex inverse hyperbolic sine" etc. Some authors use "complex area hyperbolic sine" etc.

A complex or imaginary number is infinite if one of its components is infinite, even if the other component is NaN.

A complex or imaginary number is finite if both components are neither infinities nor NaNs.

A complex or imaginary number is a zero if both components are positive or negative zeroes.

[edit] Example

#include <stdio.h>
#include <complex.h>
#include <tgmath.h>
 
int main(void)
{
    double complex z1 = I * I;     // imaginary unit squared
    printf("I * I = %.1f%+.1fi\n", creal(z1), cimag(z1));
 
    double complex z2 = pow(I, 2); // imaginary unit squared
    printf("pow(I, 2) = %.1f%+.1fi\n", creal(z2), cimag(z2));
 
    double PI = acos(-1);
    double complex z3 = exp(I * PI); // Euler's formula
    printf("exp(I*PI) = %.1f%+.1fi\n", creal(z3), cimag(z3));
 
    double complex z4 = 1+2*I, z5 = 1-2*I; // conjugates
    printf("(1+2i)*(1-2i) = %.1f%+.1fi\n", creal(z4*z5), cimag(z4*z5));
}

Output:

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

[edit] See also

C++ documentation for Complex number arithmetic