(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given that f0 = u(x,y) + i*v(x,y) is analytic in domain D, are these functions analytic in domain D?

1. f1 = (u^2 - v^2) - 2i*u*v

2. f2 = (e^u)cos(v) + i*(e^u)sin(v)

3. f3 = u - i*v

2. Relevant equations

cauchy-riemann equations

3. The attempt at a solution

I kind of have an idea but I want to double check if my reasoning is right.

If f0 is analytic then all of the partial derivatives of u(x,y) and v(x,y) wrt x and y exist and are continuous on domain D.

1. f1 = f0^2. The product of differentiable and continuous function is also differentiable and continuous on domain D. Therefore f1 is analytic on domain D.

2. f2 = e^u * 0.5*(e^-v + e^v) + i * e^u * 0.5(e^-v - e^v)

Unsure. I can't explicity use Cauchy-Riemann to check. But I know that v and u are continuous. If a function g(x) is continuous and differentiable on some domain or set, is h(x) = e^g(x) also? If so, then f2 is analytic.

3. f3 = conjugate(f0). Not differentiable because conjugate(z) is not differentiable everywhere. Therefore not analytic.

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# Homework Help: Are these functions analytic?

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