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GNU MPC is a portable library written in C for arbitrary precision arithmetic on complex numbers providing correct rounding. It implements a multiprecision equivalent of the C99 standard. It builds upon the GNU MP and the GNU MPFR libraries.
Everyone should read GNU MPC Basics. If you need to install the library yourself, you need to read Installing GNU MPC, too.
The remainder of the manual can be used for later reference, although it is probably a good idea to skim through it.