Complex number arithmetic
If the macro constant | (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 | (since C99) (until C11) |
Imaginary numbers are supported if | (since C11) |
Defined in header <complex.h> | ||
Types | ||
(C99) | imaginary type macro (keyword macro) | |
(C99) | complex type macro (keyword macro) | |
The imaginary constant | ||
(C99) | the imaginary unit constant i (macro constant) | |
(C99) | the complex unit constant i (macro constant) | |
(C99) | the complex or imaginary unit constant i (macro constant) | |
Manipulation | ||
(C11)(C11)(C11) | constructs a complex number from real and imaginary parts (function macro) | |
(C99)(C99)(C99) | computes the real part of a complex number (function) | |
(C99)(C99)(C99) | computes the imaginary part a complex number (function) | |
(C99)(C99)(C99) | computes the magnitude of a complex number (function) | |
(C99)(C99)(C99) | computes the phase angle of a complex number (function) | |
(C99)(C99)(C99) | computes the complex conjugate (function) | |
(C99)(C99)(C99) | computes the projection on Riemann sphere (function) | |
Exponential functions | ||
(C99)(C99)(C99) | computes the complex base-e exponential (function) | |
(C99)(C99)(C99) | computes the complex natural logarithm (function) | |
Power functions | ||
(C99)(C99)(C99) | computes the complex power function (function) | |
(C99)(C99)(C99) | computes the complex square root (function) | |
Trigonometric functions | ||
(C99)(C99)(C99) | computes the complex sine (function) | |
(C99)(C99)(C99) | computes the complex cosine (function) | |
(C99)(C99)(C99) | computes the complex tangent (function) | |
(C99)(C99)(C99) | computes the complex arc sine (function) | |
(C99)(C99)(C99) | computes the complex arc cosine (function) | |
(C99)(C99)(C99) | computes the complex arc tangent (function) | |
Hyperbolic functions | ||
(C99)(C99)(C99) | computes the complex hyperbolic sine (function) | |
(C99)(C99)(C99) | computes the complex hyperbolic cosine (function) | |
(C99)(C99)(C99) | computes the complex hyperbolic tangent (function) | |
(C99)(C99)(C99) | computes the complex arc hyperbolic sine (function) | |
(C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function) | |
(C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function) | |
[edit] Notes
The following function names are potentially(since C23) 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 C23), along with their -f and -l suffixed variants.
Although the C standard names the inverse hyperbolic 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 parts is infinite, even if the other part is NaN.
A complex or imaginary number is finite if both parts are neither infinities nor NaNs.
A complex or imaginary number is a zero if both parts are positive or negative zeroes.
While MSVC does provide a <complex.h> 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.
[edit] Example
#include <complex.h> #include <stdio.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] References
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[edit] See also
C++ documentation for Complex number arithmetic |
