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Morera's theorem
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(Theorem)
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Morera's theorem provides the converse of Cauchy's integral theorem.
Theorem [1] Suppose  is a region in
 , and
 is a continuous function. If for every closed triangle  in  , we have
then  is analytic on  . (Here,
 is the piecewise linear boundary of  .)
In particular, if for every rectifiable closed curve  in  , we have
 then  is analytic on  . Proofs of this can be found most undergraduate books on complex analysis [2,3].
- 1
- W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.
- 2
- E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
- 3
- R.A. Silverman, Introductory Complex Analysis, Dover Publications, 1972.
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"Morera's theorem" is owned by matte. [ full author list (3) | owner history (2) ]
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Cross-references: complex analysis, proofs, closed curve, rectifiable, piecewise, analytic, triangle, closed, continuous function, region, Cauchy's integral theorem, converse
There are 2 references to this entry.
This is version 9 of Morera's theorem, born on 2002-08-23, modified 2005-05-16.
Object id is 3339, canonical name is MorerasTheorem.
Accessed 4167 times total.
Classification:
| AMS MSC: | 30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory) |
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Pending Errata and Addenda
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