Question

Determine the analytic function f(z) = u +iv given
that 3u + 2y = y2 – x2 + 16xy​

Answer 1

Answer:

Differentiating the given relation w . r . t x and y

3∂u∂x+2∂v∂x=−2x+16y ---------- (i)

And

3∂u∂y+2∂v∂y=2y+16x ----------- (ii)

But ux = vy and uy = -vx

Hence from (ii) we get

−3∂v∂x+2∂u∂x=2y+16x -------------- (iii)

Now multiply (i) by (3) and (iii) by (2) and add

:.13∂u∂x=26x+52yi.e,∂u∂x=2x+4y=φ1(x,y)

Again , multiply (i) by (-2) and (iii) by (3) and add

:.−13∂v∂x=56x–26yi.e∂v∂x=−4x+2y=φ2(x,y)

But f1(z)=∂u∂x+i∂v∂x=φ1(x,y)+iφ2(x,y)

Now making use of Milne Thompson i.e replacing x with z and y with 0 we get = φ1(z ,0) + i φ2(z,0)

:.f1(z)=2z–i.4z

Integrating both sides we get

f(z)=2z22–i4z22+c=z2–2iz2+c=z2(1−2i)+c