Morera theorem

If a (single-valued) function of a complex variable in a domain is continuous and if its integral over any closed rectifiable contour is equal to zero, that is, if

(*)

then is an analytic function in . This theorem was obtained by G. Morera [1].

The conditions of Morera's theorem can be weakened by restricting the requirement on vanishing integrals (*) to those taken over the boundary of any triangle that is compactly contained in , i.e. such that . Morera's theorem is an (incomplete) converse of the Cauchy integral theorem and is one of the basic theorems in the theory of analytic functions.

Morera's theorem can be generalized to functions of several complex variables. Let be a function of the complex variables , , continuous in a domain of and such that its integral vanishes when taken over the boundary of any prismatic domain compactly contained in of the form

where , , , are line segments in the planes with end points and , and is a triangle in the plane . Then is a holomorphic function in .

References

[1]	G. Morera, "Un teorema fondamentale nella teorica delle funzioni di una variabili complessa" Rend. R. Ist. Lomb. Sci. Lettere , 19 (1886) pp. 304–308
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[4]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

References

[a1]	R. Remmert, "Funktionentheorie" , 1 , Springer (1984)
[a2]	J.B. Conway, "Functions of one complex variable" , Springer (1978)