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What is the difference in terms of the physical meaning of the two sets of the following Cauchy integral, $$ \begin{split} \int_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\ \int_c \frac{1}{t^k} \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=0 \end{split} $$ from G. N. SAVIN (1968), Stress Distribution around Holes, NASA Technical Translation,
with respect to this one $$ \begin{split} \int_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &= 0\\ \int_c \frac{1}{t^k} \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=\frac{4\pi i}{\zeta^k} \end{split} $$

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?

BeeTiau
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    Roughly speaking, may be in one of them $k>0$ while in the other $k<0$... – Mostafa Ayaz Sep 06 '19 at 10:08
  • Thanks. 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? – BeeTiau Sep 06 '19 at 10:12
  • Well. The reason I think, is more practical than mathematical. May be one was dealing more with negative $k$s and the other cared the positive cases. I mention, these are all just guesses. – Mostafa Ayaz Sep 06 '19 at 10:14
  • I have added another basic assumption for the equations, that in both cases $k \geq 1$ – BeeTiau Sep 06 '19 at 10:24
  • Is $C$ explicitly defined? e.g. unit circle? – Mostafa Ayaz Sep 06 '19 at 10:26
  • Yes. That's right. $C$ is a unit circle. – BeeTiau Sep 06 '19 at 10:26
  • One more question: what about $|\zeta|$? – Mostafa Ayaz Sep 06 '19 at 10:43
  • It is defined as $\zeta = e^{k \theta i} = \cos k \theta + i \sin k \theta$ – BeeTiau Sep 06 '19 at 10:45
  • If $|\zeta| < 1$ and the integral is taken in the ccw direction in the first two formulas and $|\zeta| > 1$ and the integral is taken in the cw direction in the last two formulas, then both sets of formulas are correct. If $|\zeta| = 1$, as $\zeta^k = e^{i k \theta}$ implies, then the formulas are not correct, even if you take the Cauchy principal value. – Maxim Sep 06 '19 at 13:46
  • Does it mean that the first two formulae integrate a region "inside" the circle $c$ while the last two formulae integrate a region "outside" the circle? – BeeTiau Sep 07 '19 at 09:23

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