complex mean-value theorem
Theorem [1] Suppose $\mathrm{\Omega}$ is an open convex set in $\u2102$, suppose $f$ is a holomorphic function^{} $f:\mathrm{\Omega}\to \u2102$, and suppose $a,b$ are distinct points in $\mathrm{\Omega}$. Then there exist points $u,v$ on ${L}_{ab}$ (the straight line connecting $a$ and $b$ not containing the endpoints^{}), such that
$\mathrm{\Re}\{{\displaystyle \frac{f(b)-f(a)}{b-a}}\}=\mathrm{\Re}\{{f}^{\prime}(u)\},$ | ||
$\mathrm{\Im}\{{\displaystyle \frac{f(b)-f(a)}{b-a}}\}=\mathrm{\Im}\{{f}^{\prime}(v)\},$ |
where $\mathrm{\Re}$ and $\mathrm{\Im}$ are the real (http://planetmath.org/RealPart) and imaginary parts^{} of a complex number^{}, respectively.
References
- 1 J.-Cl. Evard, F. Jafari, A Complex Rolle’s Theorem, American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861.
Title | complex mean-value theorem |
---|---|
Canonical name | ComplexMeanvalueTheorem |
Date of creation | 2013-03-22 13:49:02 |
Last modified on | 2013-03-22 13:49:02 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 7 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 26A06 |