  In classical complex analysis analyticity of a complex function $f$ is
equivalent to differentiability of its real and imaginary parts $u$ and $v$,
respectively, together with the Cauchy-Riemann equations for the partial
derivatives of $u$ and $v$. We extend this result to the context of free
noncommutative functions on tuples of matrices of arbitrary size. In this
context, the real and imaginary parts become so called real noncommutative
functions, as appeared recently in the context of L\"owner's theorem in several
noncommutative variables. Additionally, as part of our investigation of real
noncommutative functions, we show that real noncommutative functions are in
fact noncommutative functions.
