result for meromorphic functions, and a related result, the argument principle. It is known from Morera's theorem that the uniform limit of holomorphic functions is holomorphic. This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series. In addition the Cauchy formulas for the higher order derivatives show that all these derivatives also converge uniformly. The analog of the Cauchy integral formula in real analysis is the Poisson integral formula for harmonic functions; many of the results for holomorphic functions carry over to this setting. No such results, however, are valid for more general classes of differentiable or real analytic functions. For instance, the existence of the first derivative of a real function need not imply the existence of higher order derivatives, nor in particular the analyticity of the function. Likewise, the uniform limit of a sequence of (real) differentiable functions may fail to be differentiable, or may be differentiable but with a derivative which is not the limit of the derivatives of the members of the sequence. Another consequence is that if is holomorphic in and then the coefficients satisfy Cauchy's inequality From Cauchy's inequality, one can easily deduce that every bounded entire function must be constant (which is Liouville's theorem). The formula can also be used to derive Gauss's Mean-Value Theorem, which states In other words, the average value of over the circle centered at with radius is . This can be calculated directly via a parametrization of the circle. Generalizations Smooth functions A version of Cauchy's integral formula is the Cauchy–Pompeiu formula, and holds for smooth functions as well, as it is based on Stokes' theorem. Let be a disc in and suppose that is a complex-valued function on the closure of . Then One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in . Indeed, if is a function in , then a particular solution of the equation is a holomorphic function outside the support of . Moreover, if in an open set D, for some (where ), then is also in and satisfies the equation The first conclusion is, succinctly, that the convolution of a compactly supported measure with the Cauchy kernel is a holomorphic function off the support of . Here denotes the principal value. The second conclusion asserts that the