GATE 2016 EE – SET 2 – Question 6

Let $$z=x+iy$$ and $$z^{\ast }=x-iy$$

$$f(z)=x+iy+x-iy=2x$$

A function is said to be continuous at a point x=a when the following conditions are satisfied. 

(1) f(a) exists.

(2) $$\lim_{x\to a}f(x)$$ exists

(3) $$f(a)=\lim_{x\to a}f(x)$$

Therefore the function is continuous. 

Let $$f(z)=2z+i0 = u+iv$$; $$u = 2x, v = 0$$

$$\frac{\partial u}{\partial x}=2; \frac{\partial u}{\partial y}=0; \frac{\partial v}{\partial x}=0; \frac{\partial v}{\partial y}=0$$

$$\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$$

Therefore, the real and imaginary parts of f(z) do not satisfy Cauchy Riemann equations. 

This implies that the function is not analytic.