What is the difference in terms of the physical meaning ofbetween 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, 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 $\zeta > 1$?
