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I don't understand how we get the integral in line 11 of the proof. Line 11 is as follows:

$$ \frac{1}{2\pi \sqrt{-1}} \int_{\partial \Delta_{\epsilon}} f(w)\frac{dw} {(w - z)} = \int_{\partial \Delta} f(w)\frac{dw} {(w - z)} + \int_{\Delta - \Delta_{\epsilon}} \frac{\partial f(w)}{\partial \bar{w}}\frac{dw\wedge d\bar{w}} {(w - z)} $$

I know that $\int_{\Delta - \Delta_{\epsilon}} \frac{\partial f(w)}{\partial \bar{w}}\frac{dw\wedge d\bar{w}} {(w - z)} = \int_{\partial(\Delta - \Delta_{\epsilon})} f(w)\frac{dw} {(w - z)} $. What I don't understand is how those integrals add up to original one. How do we know that $\partial(\Delta_{\epsilon}) = \partial \Delta + \Delta - \Delta_{\epsilon}.$ enter image description here

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joe blacksmith
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  • Just posting large screenshots of equations like this is frowned upon; it is preferred that you include equations using MathJax. Also, which one is line 11? This is one of the reasons why we prefer that MathJax is used for equations. – The Pointer Feb 05 '21 at 22:53
  • I wrote line 11 beneath – joe blacksmith Feb 05 '21 at 22:56
  • Oh, I see. You should make it clear; see my edit. – The Pointer Feb 05 '21 at 22:58
  • Thank you for editing it. – joe blacksmith Feb 05 '21 at 22:59
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    It is $$\int_{\Delta-\Delta_\epsilon}d\eta=\int_{\partial(\Delta-\Delta_\epsilon)}\eta\ .$$ – dan_fulea Feb 05 '21 at 23:05
  • @dan_fulea Yes I deduced that as well. However that doesn't equate to $\int_{\partial \Delta_{\epsilon}}$ – joe blacksmith Feb 05 '21 at 23:10
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    Which is the boundary exactly of $\Delta-\Delta_\epsilon$? (And which orientation is used...?!) Please always write in the question what you know, this avoids such conversations and leads to a quick pointed answer for potential answerers. – dan_fulea Feb 05 '21 at 23:15
  • Ok I am posting it right now. 1 moment. – joe blacksmith Feb 05 '21 at 23:18
  • I have edited the question. – joe blacksmith Feb 05 '21 at 23:23
  • You need $\partial (\Delta -\Delta_\epsilon) = \partial \Delta - \partial \Delta_\epsilon$ – Arctic Char Feb 05 '21 at 23:28
  • @ArcticChar can you see my edit above. – joe blacksmith Feb 05 '21 at 23:47
  • I dont know what to say besides the above. Did you try to understand/verify what i and dan tried to tell you? – Arctic Char Feb 06 '21 at 02:56
  • @ArcticChar I will try explain what is exactly bugging me. Any integral can be broken down into its components. For instance $\int_{1}^3f dx = \int_{1}^2 f dx + \int_{2}^3 f dx$. This is what we are trying to do here right ? However, the subscripts don't add up to the original intergral. – joe blacksmith Feb 06 '21 at 03:07
  • @ArcticChar how do we know that $\partial(\Delta_{\epsilon}) = \partial \Delta + \partial(\Delta - \Delta_{\epsilon}).$? Shouldn't it be the other way around

    $\partial(\Delta_{\epsilon}) = \partial \Delta + \partial(\Delta_{\epsilon} - \Delta) = \partial(\Delta_{\epsilon})$?

     – joe blacksmith Feb 06 '21 at 03:10
  • @ArcticChar I understand it now it is just things were not in the right side so that is why we get that. Thank you for your help and Dan. – joe blacksmith Feb 10 '21 at 01:52

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