Morera's theorem

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 ${\displaystyle z\in D}$, 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

$${\displaystyle F(z)=\int _{\gamma }f(\zeta )\,d\zeta .}$$

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,

$${\displaystyle \int _{\gamma }f(\zeta )\,d\zeta +\int _{\tau ^{-1}}f(\zeta )\,d\zeta =\oint _{\gamma \tau ^{-1}}f(\zeta )\,d\zeta =0.}$$

And it follows that

$${\displaystyle \int _{\gamma }f(\zeta )\,d\zeta =\int _{\tau }f(\zeta )\,d\zeta .}$$

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.

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.

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

$${\displaystyle \oint _{C}f(z)\,dz=\oint _{C}\lim _{n\to \infty }f_{n}(z)\,dz=\lim _{n\to \infty }\oint _{C}f_{n}(z)\,dz=0}$$

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

$${\displaystyle \oint _{C}\Gamma (\alpha )\,d\alpha =0}$$

for a suitable closed curve C, by writing

$${\displaystyle \oint _{C}\Gamma (\alpha )\,d\alpha =\oint _{C}\int _{0}^{\infty }x^{\alpha -1}e^{-x}\,dx\,d\alpha }$$

and then using Fubini's theorem to justify changing the order of integration, getting

$${\displaystyle \int _{0}^{\infty }\oint _{C}x^{\alpha -1}e^{-x}\,d\alpha \,dx=\int _{0}^{\infty }e^{-x}\oint _{C}x^{\alpha -1}\,d\alpha \,dx.}$$

Then one uses the analyticity of α ↦ x^α−1 to conclude that

$${\displaystyle \oint _{C}x^{\alpha -1}\,d\alpha =0,}$$

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.

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.

• Cauchy–Riemann equations
• Methods of contour integration
• Residue (complex analysis)
• Mittag-Leffler's theorem

• 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.

• "Morera theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Morera's Theorem". MathWorld.