PLEASE!! can someone help me with a proof for complex variables involving line integrals?

Question

Show that if P&Q are continuous complex-valued functions on a curve γ, then F(w)=∫γ = p/(z-w) dx + ∫γ = q/(z-w) dy (where z=x+iy) is analytic for w Є C\γ. Express F’(w) as a line integral over γ.

Answer 1

Not the answer, but a suggestion.



This is a college level questionm, so I'm assuming that you are in college. Go to the math department library and find other texts on complex analysis. Look at the chapters that cover the same material. Look at the examples the authors have worked out. You may be surprised to find this problem or one enough like it to give you a clue on how to proceed.



This is not that tough a problem. N years ago, (where 0 < N < 50) I could have given you an answer, but since then, the analysis sections of my brain have shrunken.

Answer 2

What you wrote makes no experience. i'm able to make guesses about what you recommend (analyticity of particular purposes defined in words of line integrals) yet what you actually wrote is garbage. One hint for once you get your notation straightened out: differentiation below the vital signal.