Prove that the function is analytic A FzZez B same question

Prove that the function is analytic:

A) F(z)=Z(e^-z)

B) (same question) then what is the equation to find the inverse of f(z) = i/cos(z)

Please explain your steps, be clear and consistent this is advanced engineering math.
Thank you!

Solution

f(z) = ze^-z

Let z =x+iy

f(z) = (x+iy) e^-x-iy

   =(x+iy) e^-x(cosy - i siny)

= e^-x[x cosy + y sin y+i(ycosy -x sin y)

Real part u = e^-x[x cosy + y sin y]

Imaginary part v =e^-x[(ycosy -x sin y)]

u_x = e^-x[cosy]-e^-x[x cosy + y sin y] : v_y = e^-x[cosy-ysin y -x cosy]

u_y = e^-x[-x siny + sin y + y cosy]: v_x = -e^-x[(ycosy -x sin y)]+e^-x[(- sin y)]

We observe that u_x = v_y and u_y =v_x

So f is analytic.

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f(z) = i /cos z

= i sec z

z = -if(z) cos z

So inverse is -if(z) cos z