second form of Cauchy integral theorem
Theorem.
Let the complex function f be analytic in a simply connected open domain U of the complex plane, and let a and b be any two points of U.β Then the contour integral
β«Ξ³f(z)πz | (1) |
is independent on the path Ξ³ which in U goes from a to b.
Example.β Letβs consider the integral (1) of the real part function defined by
f(z):= |
with the path going from the pointβ β to the pointβ .β If is the line segment , we may use the substitution
and (1) equals
Secondly, we choose for the broken line whereβ .β Now (1) is the sum
Thus, the integral (1) of the function depends on the path between the two points.β This is explained by the fact that the function is not analytic β its real part and imaginary part 0 do not satisfy the Cauchy-Riemann equations.
Title | second form of Cauchy integral theorem |
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Canonical name | SecondFormOfCauchyIntegralTheorem |
Date of creation | 2013-03-22 15:19:39 |
Last modified on | 2013-03-22 15:19:39 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 9 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 30E20 |
Synonym | equivalent form of Cauchy integral theorem |
Related topic | CauchyIntegralTheorem |
Defines | example of non-analytic function |