variant of Cauchy integral formula
Theorem. Let be holomorphic in a domain of . If is a closed contour not intersecting itself which with its domain is contained in and if is an arbitrary point inside , then
(1) |
Proof. Let be any positive number. Denote by the circles with radius and centered in . We have
According to the corollary of Cauchy integral theorem and its example, we may write
If , we have
The continuity of in the point implies, that
when i.e. when
(2) |
If (2) is in , we obtain first
whence, by the estimation theorem of integral,
and lastly
(3) |
This result implies (1).
Title | variant of Cauchy integral formula |
Canonical name | VariantOfCauchyIntegralFormula |
Date of creation | 2013-03-22 18:54:15 |
Last modified on | 2013-03-22 18:54:15 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 6 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 30E20 |
Synonym | Cauchy integral formula |
Related topic | CauchyIntegralFormula |
Related topic | CorollaryOfCauchyIntegralTheorem |
Related topic | ExampleOfFindingTheGeneratingFunction |
Related topic | GeneratingFunctionOfLaguerrePolynomials |
Related topic | GeneratingFunctionOfHermitePolynomials |