Problem

Newton’s method can be defined for the equation f (z) = g(x, y) + ih(x, y), where f (z)...

Newton’s method can be defined for the equation f (z) = g(x, y) + ih(x, y), where f (z) is an analytic function of the complex variable z = x + iy (x and y real) and g(x, y) and h(x, y) are real functions for all x and y. The derivative f ′(z) is given by f ′(z) = g_x +ih_x = h_y−ig_y because the Cauchy-Riemann equations g_x = h_y and h_x = −g_y hold. Here the partial derivatives are defined as g_x = ∂g/∂x, g_y = ∂g/∂y, and so on. Show that

Newton’s method

can be written in the form

Here all functions are evaluated at z_n = x_n + iy_n.

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Solutions For Problems in Chapter 3.2

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