Thread: show sin(z-bar) is not differentiable everywhere

z is a complex number; z = x+iy
z-bar is a complex conjugate; z-bar = x-iy

need to show: sin(z-bar) is not differentiable everywhere

formulas (if needed):



2) Cauchy Riemann equations





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problem is i dunno how to express sin(z) as real u(x,y) + iv(x,y). I'm not even sure if C-R is the best way to go about solving this problem.

Originally Posted by rubix

z is a complex number; z = x+iy
z-bar is a complex conjugate; z-bar = x-iy

need to show: sin(z-bar) is not differentiable everywhere

formulas (if needed):



2) Cauchy Riemann equations





----

problem is i dunno how to express sin(z) as real u(x,y) + iv(x,y). I'm not even sure if C-R is the best way to go about solving this problem.

Let

Let us use the fact that and



To me this looks like we left with the form



This would plot a straight line up the imaginary axis and could be considered in the same light as a vertical asymtote. We cannot differentiate a vertical line.

I havent delt with imaginary numbers all that much (besides the basic principles involved with calculus, so perhaps a more indepth assesment is required). However, I think the above should suffice as proof enough.