First of all i'm not in the exam (i still need this answer for 6 hours me and my friend will make some book and i spot weird property in some page. And i just fell sorry if my reader spot it as false statement. My time limit is tomorrow in about 6 hours.) it's not a homework, or something for bad purpose, even not my advantage. Just ignore the comment.
If you're still not sure about my question, please give me a clever logic question about proving that my question is real for non bad purpose. For example :
If i'm in exam, i will not be able to texting in my laptop or phone.
If i'm in exam, i just need the answer for 1-2 hours.
If it's a homework, logically there is no more homework considering my uni will change semester, and my lecturer won't give trivial questions like this because of their busy schedule in the last semester.
If it's for bad purpose, i would not have thought this far.
If i'm lying about the book, i CAN E-MAIL the file of my book for you. And my name and my friends are listed there
does that make any sense? I really need this answer to correct my proof in my book
Sorry for my bad prologue
Before my main question, please take a look about this website on the second paragraph (first proof) :
I want to proof:
$$\oint_C M(x,y) \Bbb dx=-\oint_C M(x,y)$$
My proof attempt:
$$\begin{align} \oint_C M(x,y) \Bbb dx &=\int_C M(x,y) \Bbb dx\\ &=-\int_{-C} M(x,y) \Bbb dx\\ &=-\oint_{-C} M(x,y) \Bbb dx\\ &=-\oint_{C} M(x,y) \Bbb dx \end{align}$$
List of my question :
- Is that property always true?
- If it isn't, in what cases will it be true?
- Are complex analytic function under that integration always satisfy that property considering Cauchy Theorem. (Notice it's a closed integral) ($0=-0=0$)
- Please verify my proof
Somehow, i doubt with my proof. Please do a correction in specific answer please (if you have a time).
And please explain with reduce the use of complicated notations. I'm newbie.
Very kind of you if you want to help me.
Thanks in advance.