Processing math: 41%

second form of Cauchy integral theorem


Theorem.

Let the complex function f be analytic in a simply connected open domain U of the complex planeMathworldPlanetmath, and let a and b be any two points of U.  Then the contour integral

∫γf(z)𝑑z (1)

is independent on the path Ξ³ which in U goes from a to b.

Example.  Let’s consider the integral (1) of the real part function defined by

f(z):=

with the path Ξ³ going from the point  O=(0, 0)  to the point  Q=(1, 1).  If Ξ³ is the line segmentMathworldPlanetmath O⁒Q, we may use the substitution

z:=(1+i)⁒t,d⁒z=(1+i)⁒d⁒t,0≦t≦1,

and (1) equals

∫01tβ‹…(1+i)⁒𝑑t=12+12⁒i.

Secondly, we choose for Ξ³ the broken line O⁒P⁒Q where  P=(1, 0).  Now (1) is the sum

∫O⁒PRe⁒(z)⁒𝑑z+∫P⁒QRe⁒(z)⁒𝑑z=∫01x⁒𝑑x+∫01i⁒𝑑y=12+i.

Thus, the integral (1) of the function depends on the path between the two points.  This is explained by the fact that the function f is not analytic β€” its real part x and imaginary part 0 do not satisfy the Cauchy-Riemann equationsMathworldPlanetmath.

Title second form of Cauchy integral theorem
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