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Question

# An analytic function of a complex variable z=x+iy is expressed as f(z)=u(z,y)+iv(x,y), where i=√−1. if u(x,y)=x2, then expansion for v(z,y) in terms of x,y and a general constant c would be

A
xy+c
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B
x2+y22+C
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C
2xy+c
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D
(xy)22+c
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Solution

## The correct option is C 2xy+cUsing MILNE THOMSON method f(z)=u+iv where u=x2−y2 (Real part is given) Step 1: ∂u∂x=2x≈ϕ1(x,y) Step 2: ϕ1 (z,0)=2z Step 3: ∂u∂y=−2y≈ϕ2 (x,y) Step 4: ϕ2 (z,0)=0 Step 5: f(z)=∫[ϕ1(z,0)−iϕ2(z,0)]dz+c =∫(2z−0)dz+c =z2+c u+iv=x2−y2+2ixy+c Hence v=2xy+c

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