Question

if u+v=2sin2x/e^zy+e^-zy-cos2x then find an analytic function f(z)=u+iv in terms of x​

Answer 1

Answer:

Step-by-step explanation:  If u+v=2sin2xe2y+e−2y−2cos2x then find corresponding analytical function f(z)=u+iv

My attempt: we have (1+i)f(z)=u−v+i(u+v)=U+iV (say)

then

V=u+v=sin2xcosh2y−cos2x (from given equation u+v)

dU=Uxdx+Uydy=Vydx−Vxdy(from cauchy reiman equations)

Now i got

Vx=2cos2xcosh2y−2(cosh2y−cos2x)2,Vy=−2sin2xsinh2y(cosh2y−cos2x)2

dU=Vydx−Vxdy

Substituting above Vx,Vy values in dU i am not able to proceed to find U. This is getting messy. Can you provide me any hint?