Morera's theorem

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 a 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 $${\oint }_{\gamma }f(z)dz=0$$ for every closed piecewise C¹ curve $\gamma$ 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 = re^iθ. 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 the result will still be an antiderivative of 1/z.

Proof

The integrals along two paths from a to b are equal, since their difference is the integral along a closed loop.

To prove the theorem, we construct an anti-derivative for f. Since the anti-derivative is holomorphic (by construction!), and since holomorphic functions are analytic, it follows that f is holomorphic.

Without loss of generality, it can be assumed that D is connected. Fix a point z₀ in D, and for any $z\in D$, let $\gamma :[0,1]\to D$ be a piecewise C¹ curve such that $\gamma (0)=z_0$ and $\gamma (1)=z$. Then define the function F to be $$F(z)=\underset{\gamma }{\int }f(\zeta )d\zeta .$$

To see that the function is well-defined, suppose $\tau :[0,1]\to D$ is another piecewise C¹ curve such that $\tau (0)=z_0$ and $\tau (1)=z$. The curve $\gamma {\tau }^{-1}$ (i.e. the curve combining $\gamma$ with $\tau$ in reverse) is a closed piecewise C¹ curve in D. Then, $$\underset{\gamma }{\int }f(\zeta )d\zeta +\underset{{\tau }^{-1}}{\int }f(\zeta )d\zeta ={\oint }_{\gamma {\tau }^{-1}}f(\zeta )d\zeta =0.$$

And it follows that $$\underset{\gamma }{\int }f(\zeta )d\zeta =\underset{\tau }{\int }f(\zeta )d\zeta .$$

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.

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 $${\oint }_C{f}_n(z)dz=0$$ for every n, along any closed curve C in the disc. Then the uniform convergence implies that $${\oint }_{C}f(z)dz={\oint }_C\underset{n\to \mathrm{\infty }}{\lim }{f}_n(z)dz=\underset{n\to \mathrm{\infty }}{\lim }{\oint }_C{f}_n(z)dz=0$$ 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 $$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$$ or the Gamma function $$\Gamma (\alpha )=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-x}dx.$$

Specifically one shows that $${\oint }_C\Gamma (\alpha )d\alpha =0$$ for a suitable closed curve C, by writing $${\oint }_C\Gamma (\alpha )d\alpha ={\oint }_C\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-x}dxd\alpha$$ and then using Fubini's theorem to justify changing the order of integration, getting $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{\oint }_C{x}^{\alpha -1}{e}^{-x}d\alpha dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{\oint }_C{x}^{\alpha -1}d\alpha dx.$$

Then one uses the analyticity of α ↦ x^α−1 to conclude that $${\oint }_C{x}^{\alpha -1}d\alpha =0,$$ 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 $${\oint }_{\partial T}f(z)dz$$ 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

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