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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+c
Using MILNE THOMSON method
f(z)=u+iv
where u=x2y2 (Real part is given)
Step 1: ux=2xϕ1(x,y)
Step 2: ϕ1 (z,0)=2z
Step 3: uy=2yϕ2 (x,y)
Step 4: ϕ2 (z,0)=0
Step 5: f(z)=[ϕ1(z,0)iϕ2(z,0)]dz+c
=(2z0)dz+c
=z2+c
u+iv=x2y2+2ixy+c
Hence v=2xy+c

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