Morera’s theorem
Theorem [1] Suppose $G$ is a region in $\u2102$, and $f:G\to \u2102$ is a continuous function^{}. If for every closed triangle $\mathrm{\Delta}$ in $G$, we have
$${\int}_{\partial \mathrm{\Delta}}f\mathit{d}z=0,$$ |
then $f$ is analytic^{} on $G$. (Here, $\partial \mathrm{\Delta}$ is the piecewise linear boundary (http://planetmath.org/BoundaryInTopology) of $\mathrm{\Delta}$.)
In particular, if for every rectifiable closed curve $\mathrm{\Gamma}$ in $G$, we have ${\int}_{\mathrm{\Gamma}}f\mathit{d}z=0,$ then $f$ is analytic on $G$. Proofs of this can be found most undergraduate books on complex analysis [2, 3].
References
- 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.
Title | Morera’s theorem |
---|---|
Canonical name | MorerasTheorem |
Date of creation | 2013-03-22 12:58:09 |
Last modified on | 2013-03-22 12:58:09 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 12 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 30D20 |