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1)
(z^2+3)/((z+1)(z^2+5))
Product rule, chain rule, and power rule can be used to find a
derivative except where the function is undefined, (z+1)(z^2+5) = 0.
These are z = -1, \pm\sqrt{5}
2)
f(x+iy) = xy + iy
Ux = y
Uy = x
Vx = 0
Vy = i
Ux = Vy? y is a real number, so it is not equal to i. This is,
therefore, not analytic.
3)
f(z) = z^2 = x^2 + 2xyi - y^2
Ux = 2x
Uy = -2y
Vx = 2y
Vy = 2x
Ux = Vy? Yes.
Uy = -Vx? Yes.
This function is analytic everywhere on the plane.
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