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    Cauchy-Riemann equations

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    Recall that if z=x+iy then, x=(z + zbar)/2 and y=(z-zbar)/2.

    a) By formally applying the chain rule in calculus to a function F(x,y) of two real variables, derive the expression

    dF/dz=(dF/dx)(dx/dz)+(dF/dy)(dy/dz)=1/2((dF/dx)+i(dF/dy)).

    b) Define the operator d/dz=1/2((d/dz)+i(d/dy)) suggested by part (a) to show that if the first order partial derivatives of the real and imaginary components of a function f(z)=u(x,y)+iv(x,y) satisfy the Cauchy-Riemann equations then,

    df/dz=1/2[(ux-vy)+i(vx+vy)]=0.

    Thus derive the complex form df/dzbar=0 of the Cauchy-Riemann equations.

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    https://brainmass.com/math/complex-analysis/derive-complex-cauchy-riemann-equations-344063

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