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 $U$.  Then the contour integral

 $\displaystyle\int_{\gamma}f(z)\,dz$ (1)

is independent on the path $\gamma$ which in $U$ goes from $a$ to $b$.

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

 $f(z):=\mbox{Re}(z)$

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

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

and (1) equals

 $\int_{0}^{1}t\!\cdot\!(1\!+\!i)\,dt=\frac{1}{2}\!+\!\frac{1}{2}i.$

Secondly, we choose for $\gamma$ the broken line $OPQ$ where  $P=(1,\,0)$.  Now (1) is the sum

 $\int_{OP}\mbox{Re}(z)\,dz+\int_{PQ}\mbox{Re}(z)\,dz=\int_{0}^{1}x\,dx+\int_{0}% ^{1}i\,dy=\frac{1}{2}\!+\!i.$

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

Title second form of Cauchy integral theorem SecondFormOfCauchyIntegralTheorem 2013-03-22 15:19:39 2013-03-22 15:19:39 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 30E20 equivalent form of Cauchy integral theorem CauchyIntegralTheorem example of non-analytic function