If x-iy3-u-iv- then show that ux-vy-4x2-y2

Given data, (x+iy)^3=u+iv ⇒ x^3+(iy)^3+3x.iy(x+iy)=u+iv ⇒ x^3+y^3(i)^3+3x^2yi+3xy^2(i)^2=u+iv ⇒ x^3 y^3i+3x^2yi 3xy^2=u+iv ⇒(x^3 3xy^2)+i(3x^2y y^3)=u+iv C ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

If x-iy3=u+iv...

Question

$If (x - i y)^{3} = u + i v, then show that \frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2})$

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Solution

Given data, $(x + i y)^{3} = u + i v$

$\Rightarrow x^{3} + (i y)^{3} + 3 x . i y (x + i y) = u + i v$

$\Rightarrow x^{3} + y^{3} (i)^{3} + 3 x^{2} y i + 3 x y^{2} (i)^{2} = u + i v$

$\Rightarrow x^{3} - y^{3} i + 3 x^{2} y i - 3 x y^{2} = u + i v$

$\Rightarrow (x^{3} - 3 x y^{2}) + i (3 x^{2} y - y^{3}) = u + i v$

Comparison
on equating real and imaginary parts, we get

$u = x^{3} - 3 x y^{2} and v = 3 x^{2} y - y^{3}$

$\frac{u}{x} + \frac{v}{y} = \frac{x^{3} - 3 x y^{2}}{x} + \frac{3 x^{2} y - y^{3}}{y}$

$= x^{2} - 3 y^{2} + 3 x^{2} - y^{2}$

$= 4 x^{2} - 4 y^{2}$

$= 4 (x^{2} - y^{2})$

$∴ \frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2})$

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Integration by Partial Fractions

Standard XII Mathematics

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If (x−iy)3=u+iv, then show that ux+vy=4(x2−y2)

$If (x - i y)^{3} = u + i v, then show that \frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2})$