Thanks, user131781 and Alexandre Eremenko for your kind help! unfortunately I can only accept one answer but both of your insights are valuable! It would be difficult for a person like me to find a direction and guidance from experts in here if a post such as mine above is judged as inappropriate/off-topic at the first instance without giving even a small chance to hear for one or two responses. The intention of a question is not just to find a right/wrong answer, but to seek possibilities. If you cannot give an answer/guidance, at least help us by giving a little bit more time. Thanks.

Thanks you! I really appreciate your suggestion and effort. Many people have now commented that this kind of question is not appropriate to be asked in here. I don't know why. While your response seems to be what I am looking for; a kind of direction from a math expert to someone who's completely clueless. I appreciate your effort! thanks!

Ah, this is it. If I may ask you a follow-up question. Why does $\zeta$ must be inside, i.e. $\zeta < 1$ for the first formulae to be correct? I really want to know the answer of this--Or, $\zeta > 1$ for the second formulae to be correct.

Thanks for replying my question. When you say that the books might perhaps use a "different" sense, is it something to do with the integration of the "internal region" or "external region" around the boundary $c$?

Glad to know that now you understand my intention that this is not about getting a right/wrong answer but more to get an insight.

That's okay. But I do appreciate Alexandre Eremenko's effort in trying to help me. Perhaps this is not about getting the right/wrong answer because, as I said, both formulae are correct. What I am after is an insight from someone like Alexandre Eremenko when he said that "I suppose that your books use the different sense of understanding these integrals."

Oh okay. I am asking this at the wrong place then, i.e. Mathematics cannot provide meaning for these mathematical expressions, or how to make sense of those mathematical formulae. Both formulae are actually relevant in my case; just that I could not understand why the two authors came up with an 'opposite' relations in their formulation. Unfortunately, both of them don't provide sufficient explanation as to why they used that formula. I was hoping that mathematician could help me to provide the best possible explanation. But I am bit unlucky now.

This is about mathematic. I am trying to understand the meaning of this mathematical expression. Thanks!