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In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped.

I personally interacted with many people to understand the meaning of complex differentials or complex differentials form, but no one explained in intuitive or motivational way what that means.

I tried to follow some books like -

  • Calculus on manifolds- Spivak,

  • Principles of Mathematical Analysis - Rudin,

  • Analysis on Manifolds - Munkres,

or even some specifically written books on differential forms like

  • differential forms - Weintraub

Everyone tried to explain it through algebraic meanings, but that is too abstract; I can not see its relations with analysis quickly.

Question: Can one give an intuitive explanation of complex differential forms? What should we keep in mind when we came across it in some study? I would mostly welcome explanation through examples as well.

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  • $\begingroup$ I mean the fundamental idea is as far as i know that you can describe a complex space (manifold) by complex (holomorphic) coordinates and their conjugates. Then you build 1,2,3-forms out of them. What is difficult to grasp for you? $\endgroup$ – tired Oct 5 '15 at 13:49
  • $\begingroup$ For starters, do you feel you understand the role of one-forms in complex analysis, e.g., the Cauchy integral formula and residues? $\endgroup$ – Andrew D. Hwang Nov 22 '15 at 0:49

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