An analytic function of a complex variable z-x-iy is expressed as fz-uz-y-ivx-y- where i-1- if ux-y-x2- then expansion for vz-y in terms of x-y and a general constant c would be

Using MILNE THOMSON method f(z)=u+iv where u=x^2 y^2 nbsp; (Real part is given) Step 1: ∂ u∂ x=2x≈ϕ_1(x,y) Step 2: ϕ_1 (z,0)=2 z Step 3: ∂ u∂ y= 2y≈ϕ_2 (x,y) ...

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Standard XII

Mathematics

Continuity of a Function

An analytic f...

Question

An analytic function of a complex variable $z = x + i y$ is expressed as $f (z) = u (z, y) + i v (x, y),$ where $i = \sqrt{- 1}$ . if $u (x, y) = x^{2},$ then expansion for $v (z, y)$ in terms of $x, y$ and a general constant $c$ would be

x y + c

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\frac{x^{2} + y^{2}}{2} + C

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2 x y + c

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\frac{(x - y)^{2}}{2} + c

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Solution

The correct option is C $2 x y + c$
Using MILNE THOMSON method
$f (z) = u + i v$
where $u = x^{2} - y^{2}$ (Real part is given)
Step 1: $\frac{\partial u}{\partial x} = 2 x \approx ϕ_{1} (x, y)$
Step 2: $ϕ_{1} (z, 0) = 2 z$
Step 3: $\frac{\partial u}{\partial y} = - 2 y \approx ϕ_{2} (x, y)$
Step 4: $ϕ_{2} (z, 0) = 0$
Step 5: $f (z) = \int [ϕ_{1} (z, 0) - i ϕ_{2} (z, 0)] d z + c$
$= \int (2 z - 0) d z + c$
$= z^{2} + c$
$u + i v = x^{2} - y^{2} + 2 i x y + c$
Hence $v = 2 x y + c$

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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

An analytic function of a complex variable $z = x + i y$ is expressed as $f (z) = u (z, y) + i v (x, y),$ where $i = \sqrt{- 1}$ . if $u (x, y) = x^{2},$ then expansion for $v (z, y)$ in terms of $x, y$ and a general constant $c$ would be