Discussion
Let \(\mathbb{C} = \{z = x + iy : x\) and \(y\) are real numbers, \(i = \sqrt{-1}\}\) be the set of complex numbers. Let the function \(f(z) = u(x, y) + i\, v(x, y)\) for \(z = x + iy \in \mathbb{C}\) be analytic in \(\mathbb{C}\), where
\[ u(x, y) = x\,y^{3} - y\,x^{3} \]
and
\[ \begin{aligned} v(x, y) &= \frac{x^{4}}{4} + \frac{y^{4}}{4} \\ &\quad - \frac{3}{2}x^{2}y^{2}. \end{aligned} \]
If \(f'(z)\) denotes the derivative of \(f(z)\), then
Answer: C
\(|f'(-1 + i)|^{2} = 8\).
For an analytic function \(f'(z) = u_{x} + i\,v_{x}\). From the given parts:
\[ \begin{aligned} u_{x} &= y^{3} - 3x^{2}y \\ v_{x} &= x^{3} - 3xy^{2} \end{aligned} \]
(The Cauchy-Riemann equation \(u_{x} = v_{y}\) checks: \(v_{y} = y^{3} - 3x^{2}y\).) At \(x = -1\), \(y = 1\):
\[ \begin{aligned} u_{x} &= 1 - 3 = -2,\quad v_{x} = -1 + 3 = 2 \\ |f'|^{2} &= (-2)^{2} + 2^{2} = 8 \end{aligned} \]
The values 1, 7 and 10 come from sign slips in these two derivatives.
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