Cauchy-Riemann Equations test problem

[Guest]

[tex]\lim_{x \to 0}x[/tex]Can you please tell me or guide me on how to do these question
I try to do it and I think choice D is the right answer.
Is it correct or not ?

[shyamjayakannan]

[tex]f(x+iy)=e^x(y+\cos y)+ie^x\sin y=u(x,y)+iv(x,y)[/tex] and we need to check if [tex]\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}[/tex] and [tex]\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}[/tex].

Now, [tex]\frac{\partial u}{\partial x}=e^x(y+\cos y),\frac{\partial u}{\partial y}=e^x(1-\sin y),\frac{\partial v}{\partial x}=e^x\sin y,\frac{\partial v}{\partial y}=e^x\cos y[/tex].

So, you are correct that the answer is option D.