following inversion theorems can be used. Theorem. If the characteristic function φX of a random variable X is integrable, then FX is absolutely continuous, and therefore X has a probability density function. In the univariate case (i.e. when X is scalar-valued) the density function is given by In the multivariate case it is where is the dot product. The density function is the Radon–Nikodym derivative of the distribution μX with respect to the Lebesgue measure λ: Theorem (Lévy). If φX is characteristic function of distribution function FX, two points a < b are such that is a continuity set of μX (in the univariate case this condition is equivalent to continuity of FX at points a and b), then If X is scalar: This formula can be re-stated in a form more convenient for numerical computation as For a random variable bounded from below one can obtain by taking such that Otherwise, if a random variable is not bounded from below, the limit for gives , but is numerically impractical. If X is a vector random variable: Theorem. If a is (possibly) an atom of X (in the univariate case this means a point of discontinuity of FX ) then If X is scalar: If X is a vector random variable: Theorem (Gil-Pelaez). For a univariate random variable X, if x is a continuity point of FX then where the imaginary part of a complex number is given by . And its density function is: The integral may be not Lebesgue-integrable; for example, when X is the discrete random variable that is always 0, it becomes the Dirichlet integral. Inversion formulas for multivariate distributions are available. Criteria for characteristic functions The set of all characteristic functions is closed under certain operations: A convex linear combination (with ) of a finite or a countable number of characteristic functions is also a characteristic function. The product of a finite number of characteristic functions is also a characteristic function. The same holds for an infinite product provided that it converges to a function continuous at the origin. If φ is a characteristic function and α is a real number, then , Re(φ), |φ|2, and φ(αt) are also characteristic functions. It is well known that any non-decreasing càdlàg function F with limits F(−∞) = 0, F(+∞) = 1 corresponds to a cumulative distribution function of some random variable. There is also interest in finding