Use Cauchy-Riemann equations to show that the complex function f(z) = f(x + iy) = z(x...

Use Cauchy-Riemann equations to show that the complex function f(z) = f(x + iy) = z(x + iy) is nowhere differentiable except at the origin z = 0.6 points) 2. Use Cauchy's theorem to evaluate the complex integral ekz -dz, k E R. Use this result to prove the identity 0"ck cos θ sin(k sin θ)de = 0

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Expert Solution

Answer-1. The complex function

is differentiable every where. We use the Cauchy-Riemann equations to show this. For, we write

where

Now, we compute

and hence it follows that

and hence satisfies the Cauchy-Riemann equations at all the points . Thus, it follows that is differentiable at all the points .

Colby Messinger answered 1 year ago

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