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Prove. Complex form of Green’s theorem. Let B(z, ole.gif ) be continuous and have continuous partial derivatives in a region R and on its boundary C. Then


             ole1.gif


where dA represents the element of area dxdy and z and ole2.gif are complex conjugate coordinates z = x + iy and ole3.gif = x - iy.


Proof. Let B(z, ole4.gif ) = P(x, y) + iQ(x, y) [i.e. P(x, y) + iQ(x, y) is the function obtained by substituting z = x + iy and ole5.gif = x - iy into B(z, ole6.gif ) ]. Given a region R with boundary C and a function B(z, ole7.gif ), continuous and with continuous partial derivatives in R , the integral around C is given by


ole8.gif


Applying Green’s theorem to the right member we get


ole9.gif



             ole10.gif


Now


ole11.gif


so we can write 2) as



ole12.gif


Remembering that


             ole13.gif


4) becomes


ole14.gif


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