Question

find the analytic function whose imaginary part is -sinxsinhy

Answer 1

Since ∂v/∂y=4x+2, v(x,y)=4xy+2y+ϕ(x) where ϕ(x) is an arbitrary function of x (since partial differentiation with respect to y treats x as a constant, the derivative of any function of x only, with respect to y, is 0).

From that ∂v∂x=4y+ϕ′(x)=4y so ϕ′(x)=0 and ϕ(x) is a constant.

That is, v=4xy+2y+C where C can be any constant.

Since f(x, y)= u(x,y)+ iv(x,y), f(x,y)=2x2+2x−2y2+1+(4xy+2y+C)i.

Answer 2

Answer:

Please refer the image