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**chisigma** A fundamental theorem due to the French mathematician Augustin Cauchy extablishes that if $\displaystyle f(z)$ is analytic in a region inside a closed path $\displaystyle \gamma$ is...

$\displaystyle \displaystyle \int_{\gamma} f(z)\ dz =0$ (1)

Now if $\displaystyle f(z)$ is analytic in the whole complex plane [i.e. is an *entire function*...] the (1) is valid independently from $\displaystyle \gamma$ , and that means that, if we devide $\displaystyle \gamma$ in two parts, one coming from $\displaystyle z_{1}$ to $\displaystyle z_{2}$ and the other coming back from $\displaystyle z_{2}$ to $\displaystyle z_{1}$ is...

$\displaystyle \displaystyle \int_{z_{1}}^{z_{2}} f(z)\ dz + \int_{z_{2}}^{z_{1}} f(z)\ dz =0$ (2)

... and that means that $\displaystyle \displaystyle \int_{z_{1}}^{z_{2}} f(z)\ dz $ depends only from $\displaystyle z_{1}$ and $\displaystyle z_{2}$ and *doesn't depend* from the path connecting $\displaystyle z_{1}$ and $\displaystyle z_{2}$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$