In Problem, (a) use Theorem to show that the given function is analytic in an appropriate...

In Problem, (a) use Theorem to show that the given function is analytic in an appropriate domain, and (b) use (2) or (3) to find the derivative of the function in the domain.

THEOREM Criterion for Analyticity

Suppose the real functions u(x, y) and v(x, y) are continuous and have continuous first-order partial derivatives in a domain D. If u and v satisfy the Cauchy- Riemann equations (1) at all points of D, then the complex function f(z) = u(x, y) + iv(x, y) is analytic in D.

 (1)

 (2)

 (3)