PlanetMath: second form of Cauchy integral theorem

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second form of Cauchy integral theorem

(Theorem)

Theorem 1 Let the complex function

be analytic in a simply connected open domain

of the complex plane, and let

and

be any two points of

. Then the contour integral

$\displaystyle \int_\gamma f(z)\,dz$

(1)

is independent on the path $\gamma$ which in

goes from

to

.

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

$\displaystyle f(z) :=$ Re $\displaystyle (z)$

with the path $\gamma$ going from the point $O = (0,\,0)$ to the point $Q = (1,\,1)$ . If $\gamma$ is the line segment

, we may use the substitution

$\displaystyle z := (1\!+\!i)t,\quad dz = (1\!+\!i)\,dt,\quad 0 \leqq t \leqq 1,$

and (1) equals

$\displaystyle \int_0^1t\!\cdot\!(1\!+\!i)\,dt = \frac{1}{2}\!+\!\frac{1}{2}i.$

Secondly, we choose for $\gamma$ the broken line

where $P = (1,\,0)$ . Now (1) is the sum

$\displaystyle \int_{OP}$ Re $\displaystyle (z)\,dz+\int_{PQ}$ Re $\displaystyle (z)\,dz = \int_0^1x\,dx+\int_0^1i\,dy = \frac{1}{2}\!+\!i.$

Thus, the integral (1) of the function depends on the path between the two points. This is explained by the fact that the real part function

is not analytic -- its real part

and imaginary part 0 do not satisfy the Cauchy-Riemann equations.

"second form of Cauchy integral theorem" is owned by pahio.

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See Also: Cauchy integral theorem

Other names:

equivalent form of Cauchy integral theorem

Also defines:

example of non-analytic function

Keywords:

analytic, holomorphic

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Cross-references: Cauchy-Riemann equations, imaginary part, sum, broken line, line segment, function, real part, integral, path, independent, contour integral, points, complex plane, domain, open, simply connected, analytic, complex function
There is 1 reference to this entry.

This is version 6 of second form of Cauchy integral theorem, born on 2005-06-03, modified 2008-01-03.
Object id is 7139, canonical name is SecondFormOfCauchyIntegralTheorem.
Accessed 3077 times total.

Classification:

30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

Pending Errata and Addenda

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