formula_1 is real smooth and for every compact set formula_29 there exists a constant formula_30 such that for every formula_31 and every non-negative integer formula_32 the following bound holds

Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized.For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the Fourier–Bros–Iagolnitzer transform.