Question

14. Determine at which points the function is differentiable and at which points it is analytic.

This is a problem for Complex Analysis, please answer fully, label steps, and write neatly, preferably no cursive please!

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Answer 1

G(z) = (8x - x^3 - xy^2) + i(x^2y + y^3 - 8y) u(x, y) v(x, y) u_x = d/dx (8x - x^3 - xy^2) = 8 - 3x^2 - y^2 v_x = d/dx(x^2y + y^3 - 8y) = 2xy v_y = d/dx (8x - x^3 - xy2) = -2xy v_y = d/dx (x^2y + y^3 - 8y) = x^2 + 3y^2 - 8 Now from caucnr Remain equation v_y = -2xy = -vx ux = vy rightarrow 8 - 3x^2 - y^2 = x^2 + 3y^2 - 8 rightarrow x^2 + y^2 + 3y^2 + 3x^2 - 8 - 8 = 0 rightarrow 4x^2 + 4y^2 - 16 = 0 rightarrow x^2 + y^2 - 4 = 0 rightarrow x^2 + y^2 = 4 Thus, C. R equation satisfies only if x^2 + y^2 = 4 Rightarrow for x^2 + y^2 notequalto 4 f(z) is neither dift nor Analytic Now for x^2 + y^2 = 4 we have u_x = 8 - 3x^2 - b^2 v_y = -2xy v_x = 2xy v_y = x^2 + 3y^2 - 8 All partial derivative exist and are continuous (As All are polynomials) in the nbd of x^2 + y^2 = 4 so f is dift at x^2 + y^2 = 4