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Cauchy Riemann Equation

Let f:UC function which is differentiable and U is open in C. Suppose f(z0) exists where z0=a+ib  UC. f(z)=u+iv where u:UR and v:UR. First take h=tR.f(z0)=limt0f(a+t+ib)f(a+ib)tBreaking it we get limt0u(a+t,b)u(a,b)t+ilimt0v(a+t,b)v(a,b)t=ux|z0+ivx|z0

Now take h=it, tR

f(z0)=limt0f(a+ib+it)f(a+ib)it

Breaking it down we get limt0u(a,b+t)u(a,b)it+ilimt0v(a,b+t)v(a,b)it=vy|z0iuy|z0

Equating the two equations we get f is complex differentiable at z0 and ux|z0=vy|z0vx|z0=uy|z0These two equations are called Cauchy Riemann Equation


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