Which of the following is an analytic function ?
\(z^{2}\)
A function \(f(z)=u(x,y)+iv(x,y)\) is analytic in a region when it depends on \(z\) alone (not on \(\bar{z}\)) and its real and imaginary parts satisfy the Cauchy–Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\).
For \(z^{2}=(x+iy)^{2}\) we get \(u=x^{2}-y^{2}\) and \(v=2xy\), so \(u_{x}=2x=v_{y}\) and \(u_{y}=-2y=-v_{x}\) at every point. All partial derivatives are continuous, so \(z^{2}\) is analytic in the whole plane.
For \(z+\bar{z}=2x\): \(u=2x\), \(v=0\), so \(u_{x}=2\ne v_{y}=0\); the Cauchy–Riemann equations fail everywhere.
For \(z\bar{z}=x^{2}+y^{2}\): \(u_{x}=2x\) and \(v_{y}=0\) agree only at the origin, so the function is differentiable at a single point and analytic nowhere.
For \(z/\bar{z}=e^{2i\theta}\) the value depends on the direction of approach and the function contains \(\bar{z}\) explicitly, so \(\partial f/\partial\bar{z}\ne 0\) and it is not analytic.
Check: \(\partial(z^{2})/\partial\bar{z}=0\) while \(\partial(z+\bar{z})/\partial\bar{z}=1\), \(\partial(z\bar{z})/\partial\bar{z}=z\) and \(\partial(z/\bar{z})/\partial\bar{z}=-z/\bar{z}^{2}\), confirming that only the squared function is independent of the conjugate.
Hence, \(z^{2}\).
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
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