Morera's theorem

Morera's theorem Mathematical analysis → Complex analysis Complex analysis Complex numbers Real numberImaginary numberComplex planeComplex conjugateUnit complex number Complex functions Complex-valued functionAnalytic functionHolomorphic functionCauchy–Riemann equationsFormal power series Basic Theory Zeros and polesCauchy's integral theoremLocal primitiveCauchy's integral formulaWinding numberLaurent seriesIsolated singularityResidue theoremConformal mapSchwarz lemmaHarmonic functionLaplace's equation Geometric function theory People Augustin-Louis CauchyLeonhard EulerCarl Friedrich GaussJacques HadamardKiyoshi OkaBernhard RiemannKarl Weierstrass  Mathematics portal vte If the integral along every C is zero, then f is holomorphic on D.

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies {displaystyle oint _{gamma }f(z),dz=0} for every closed piecewise C1 curve {displaystyle gamma } in D must be holomorphic on D.

The assumption of Morera's theorem is equivalent to f having an antiderivative on D.

The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.

The standard counterexample is the function f(z) = 1/z, which is holomorphic on C − {0}. On any simply connected neighborhood U in C − {0}, 1/z has an antiderivative defined by L(z) = ln(r) + iθ, where z = reiθ. Because of the ambiguity of θ up to the addition of any integer multiple of 2π, any continuous choice of θ on U will suffice to define an antiderivative of 1/z on U. (It is the fact that θ cannot be defined continuously on a simple closed curve containing the origin in its interior that is the root of why 1/z has no antiderivative on its entire domain C − {0}.) And because the derivative of an additive constant is 0, any constant may be added to the antiderivative and it's still an antiderivative of 1/z.

In a certain sense, the 1/z counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/z itself does not have an antiderivative on C − {0}.

Contents 1 Proof 2 Applications 2.1 Uniform limits 2.2 Infinite sums and integrals 3 Weakening of hypotheses 4 See also 5 References 6 External links Proof The integrals along two paths from a to b are equal, since their difference is the integral along a closed loop.

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for f explicitly.

Without loss of generality, it can be assumed that D is connected. Fix a point z0 in D, and for any {displaystyle zin D} , let {displaystyle gamma :[0,1]to D} be a piecewise C1 curve such that {displaystyle gamma (0)=z_{0}} and {displaystyle gamma (1)=z} . Then define the function F to be {displaystyle F(z)=int _{gamma }f(zeta ),dzeta .} To see that the function is well-defined, suppose {displaystyle tau :[0,1]to D} is another piecewise C1 curve such that {displaystyle tau (0)=z_{0}} and {displaystyle tau (1)=z} . The curve {displaystyle gamma tau ^{-1}} (i.e. the curve combining {displaystyle gamma } with {displaystyle tau } in reverse) is a closed piecewise C1 curve in D. Then, {displaystyle int _{gamma }f(zeta ),dzeta +int _{tau ^{-1}}f(zeta ),dzeta =oint _{gamma tau ^{-1}}f(zeta ),dzeta =0.} And it follows that {displaystyle int _{gamma }f(zeta ),dzeta =int _{tau }f(zeta ),dzeta .} Then using the continuity of f to estimate difference quotients, we get that F′(z) = f(z). Had we chosen a different z0 in D, F would change by a constant: namely, the result of integrating f along any piecewise regular curve between the new z0 and the old, and this does not change the derivative.

Since f is the derivative of the holomorphic function F, it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent power series, and the fact that power series may be differentiated term by term. This completes the proof.

Applications Morera's theorem is a standard tool in complex analysis. It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.

Uniform limits For example, suppose that f1, f2, ... is a sequence of holomorphic functions, converging uniformly to a continuous function f on an open disc. By Cauchy's theorem, we know that {displaystyle oint _{C}f_{n}(z),dz=0} for every n, along any closed curve C in the disc. Then the uniform convergence implies that {displaystyle oint _{C}f(z),dz=oint _{C}lim _{nto infty }f_{n}(z),dz=lim _{nto infty }oint _{C}f_{n}(z),dz=0} for every closed curve C, and therefore by Morera's theorem f must be holomorphic. This fact can be used to show that, for any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm.

Infinite sums and integrals Morera's theorem can also be used in conjunction with Fubini's theorem and the Weierstrass M-test to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function {displaystyle zeta (s)=sum _{n=1}^{infty }{frac {1}{n^{s}}}} or the Gamma function {displaystyle Gamma (alpha )=int _{0}^{infty }x^{alpha -1}e^{-x},dx.} Specifically one shows that {displaystyle oint _{C}Gamma (alpha ),dalpha =0} for a suitable closed curve C, by writing {displaystyle oint _{C}Gamma (alpha ),dalpha =oint _{C}int _{0}^{infty }x^{alpha -1}e^{-x},dx,dalpha } and then using Fubini's theorem to justify changing the order of integration, getting {displaystyle int _{0}^{infty }oint _{C}x^{alpha -1}e^{-x},dalpha ,dx=int _{0}^{infty }e^{-x}oint _{C}x^{alpha -1},dalpha ,dx.} Then one uses the analyticity of α ↦ xα−1 to conclude that {displaystyle oint _{C}x^{alpha -1},dalpha =0,} and hence the double integral above is 0. Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum.

Weakening of hypotheses The hypotheses of Morera's theorem can be weakened considerably. In particular, it suffices for the integral {displaystyle oint _{partial T}f(z),dz} to be zero for every closed (solid) triangle T contained in the region D. This in fact characterizes holomorphy, i.e. f is holomorphic on D if and only if the above conditions hold. It also implies the following generalisation of the aforementioned fact about uniform limits of holomorphic functions: if f1, f2, ... is a sequence of holomorphic functions defined on an open set Ω ⊆ C that converges to a function f uniformly on compact subsets of Ω, then f is holomorphic.

See also Cauchy–Riemann equations Methods of contour integration Residue (complex analysis) Mittag-Leffler's theorem References Ahlfors, Lars (January 1, 1979), Complex Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, ISBN 978-0-07-000657-7, Zbl 0395.30001. Conway, John B. (1973), Functions of One Complex Variable I, Graduate Texts in Mathematics, vol. 11, Springer Verlag, ISBN 978-3-540-90328-4, Zbl 0277.30001. Greene, Robert E.; Krantz, Steven G. (2006), Function Theory of One Complex Variable, Graduate Studies in Mathematics, vol. 40, American Mathematical Society, ISBN 0-8218-3962-4 Morera, Giacinto (1886), "Un teorema fondamentale nella teorica delle funzioni di una variabile complessa", Rendiconti del Reale Instituto Lombardo di Scienze e Lettere (in Italian), 19 (2): 304–307, JFM 18.0338.02. Rudin, Walter (1987) [1966], Real and Complex Analysis (3rd ed.), McGraw-Hill, pp. xiv+416, ISBN 978-0-07-054234-1, Zbl 0925.00005. External links "Morera theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Morera's Theorem". MathWorld. Categories: Theorems in complex analysis