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Prove that $f(z) = x^2 - y^2 + 2ixy$ is analytic and find $f'(z)$.
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written 7.8 years ago by
teamques10
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• modified 7.8 years ago |
$$u = x^2 - y^2 \hspace{0.5cm} v= 2xy$$ $$\frac{\partial u}{\partial x} = 2x \hspace{1cm} \frac{\partial u}{\partial y} = -2y$$ $$\frac{\partial v}{\partial x} = 2y \hspace{1cm} \frac{\partial v}{\partial y} = 2x$$ $$\therefore ux = vy \hspace{1cm} uy = -vx$$
Since, C-R equations are satisfied.
Hence, f(z) is analytic.
f'(z) $= ux + ivx \\ = 2x + i2y \\ = 2(x + iy) \\ = 2z$
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