Question

If F (z) = u + iv is analytic function find f (z), if u = ex ( x cosy – y sin y)

Answer 1

Answer:

It is F(z) = u + iv, F(z)cc = u - iv . u =( F + Fcc )/2 , v =-i ( F - Fcc )/2 . Follows

u - v =Re[(1+i)F(z)] = Re[(1+i)(u+iv)].. It is

(1+i)(u+iv) = (u-v) +i( u + v).

Take U =u-v = e^x(cosy-siny) , V = u+v . Both function satisfy the Cauchy- Riemann equations

U_x = V_y (1)

U_y = -V_x (2)

From (1) on has U_x =e^x(cosy -siny) = V_y . Integrating give

V = e^x(siny + cosy) + H(x). From (2)

U_y = -e^x(siny + cosY) = -V_x , obtain H’(x)=0 i.e. H(x) =C Then

V = u + v =e^x(siny + cosy) +C . Combining with

u-v = e^x(cosy - siny) obtain

F(z) = u + iv = e^x(cosy +isiny) +k = e^z +k ,k complex constant.