Question

If real part of analytic function f(z) is em cosy then f(z)=
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Answer 1

Answer:

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:

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