What is the difference between the two sets of the following Cauchy integral,
$$
\begin{split}
\int\_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\[8pt]
\int\_c \frac{1}{t^k} \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=0
\end{split}
$$
from G. N. SAVIN (1968), *[Stress Distribution around Holes](https://archive.org/details/nasa_techdoc_19710000647)*, NASA Technical Translation,  

with respect to this one
$$
\begin{split}
\int\_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &= 0\\[8pt]
\int\_c \frac{1}{t^k} \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=\frac{4\pi i}{\zeta^k}
\end{split}
$$


from V. G. UKADGAONKER and V. KAKHANDKI 2005. "Stress analysis for an orthotropic plate with an irregular shaped hole for different in-plane loading conditions—Part 1". Composite Structures, 70, 255-274.


In both cases $k \geq 1$ and $c$ is a unit circle. Also, in both cases, $\zeta^k = e^{k \theta i} = \cos k \theta + i \sin k \theta$


What consideration does each author perhaps used so that they came up with a slightly "opposite" relations? Is it something to do with the "internal region" or "external region" integration around the boundary? Why does $\zeta$ must be inside, i.e. $\zeta < 1$ for the first formulae to be correct, and for the second formulae the $\zeta > 1$?



Cauchy integral and residue theorem
A modest proposal. Could it be that there is a typo or misunderstanding in the formulation? If $\zeta$ were inside, rather than on, the circle, the first pair of formulae would be correct—-a direct consequence of the Cauchy integral formula. Analogously for the second one if it were outside.


If $\zeta$ is really on the circle, then the integrals would exist in the distributional sense—-just regard the integrand, in the usual way, as a periodic distribution on the line, but there would be no ambiguity about its possible value. I have made no attempt to actually compute it.


