Cauchy integral and residue theorem [closed]

Question

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, 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 $\zeta > 1$?

Answer 1

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.

Answer 2

Strictly speaking both formulas make no sense since all integrals are divergent because the integrands have a pole on the unit circle, namely at $\zeta$.

There are several different ways to make them meaningful: for example to bypass the singularity by a small arc inside the circle or outside the circle, or understand them as principal values, etc.) And I suppose that your books use the different sense of understanding these integrals.