Morera's theorem

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Here we follow standard texts, such as Spiegel (1964)[1] or Levinson and Redheffer (1970). [2]

If for every closed contour within a region of the complex and

then is analytic everywhere in .


Proof:

Suppose that the hypothesis is false, and that in some portion of , there exists some contour such that


where is nonzero.

We assume that the real and imaginary parts of are continuously differentiable in , (otherwise would not exist at some points in and would not be analytic by the definition of analyticity.)

Rewriting the integral by subsituting real and imaginary parts and we obtain


.


Because the partial derivatives of and exist, we may invoke the 2D version of Green's theorem for the real and imaginary parts


Because at least one of the Cauchy-Riemann equations is not satisfied, and the theorem is proven. However, this proof is less satisfactory, because we had to assume that the real and imaginary parts of were differentiable.


Alternate proof:

In general, Morera's theorem is a statement that if is continuous, then it has an anti-derivative , which is an analytic function for all in the region . Furthermore, this forces to be analytic, as well.

Suppose that we define a function and form the derivative of by the formal definition of a differentiation


.


This expression should vanish in the limit, if is the anti-derivative of .

We can rewrite this limit as

.


Because is assumed to be continuous, by the formal definition of continuity, whenever there exists, for all a such that .

Given this is true, allowing us to estimate the bound on the integrand with

.

Because is arbitrary . is analytic by the definition of analyticity as the existence of the derivative. Furthermore, because is defined by an integral, this result holds for any integration path in we have

for arbitrary and in , which in turn shows that must be analytic for all in , proving Morera's theorem.

Further ramifications of this second proof

It is possible to show from the Cauchy-Riemann equations that the derivative of an analytic function is, itself, analytic, and is infinitely differentiable in its region of analyticity. It is furthermore possible to construct a new function that is the anti-derivative of . This process may also be repeated ad infinitum.

References

[3]
[4]

[5]

  1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
  2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
  3. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
  4. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
  5. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.