Given that f(z) = SinZ, where Z=x+jy. express f(z) in the form u+jv; (ii) prove that u and v satisfy Cauchy-Riemann equatic A transformation is defined by W = (z+2j)/(z-j) find the image of the circle |Z|=2 in the w-plane and sketch it.

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Asked on March 18, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-18T05:54:19.205Z

(a)(i) Step 1: Recall the formula for z where z = x + iy. z = (x + iy) = x y + i x y. Thus, u(x,y) = x y, v(x,y) = x y. (a)(ii) Step 1: Compute partial derivatives of u. u_x = x y, u_y = x y. Step 2: Compute partial derivatives of v. v_x = - x y, v_y = x y. Step 3: Verify Cauchy-Riemann equations u_x = v_y and u_y = -v_x. u_x = x y = v_y, u_y = x y = - (- x y) = -v_x. Both hold, so f(z) is analytic. (b) Step 1: Let = z^2. For |z| = 1, || = |z|^2 = 1. The map is w = ()/( - 2). Step 2: Solve for in terms of w. = (2w)/(1 - w). Step 3: Impose || = 1. | (2w)/(1 - w) | = 1 2 |w| = |1 - w|. Step 4: Square both sides. Let w = u + iv. 4(u^2 + v^2) = (1 - u)^2 + v^2 = 1 - 2u + u^2 + v^2. Step 5: Simplify. 4u^2 + 4v^2 = 1 - 2u + u^2 + v^2 3u^2 + 3v^2 + 2u - 1 = 0, u^2 + v^2 + (2)/(3)u = (1)/(3). Step 6: Complete the square. (u + (1)/(3))^2 - ((1)/(3))^2 + v^2 = (1)/(3), (u + (1)/(3))^2 + v^2 = (1)/(3) + (1)/(9) = (4)/(9). The image is the circle |w + (1)/(3)| = (2)/(3) (center -(1)/(3), radius (2)/(3)). Sketch: Circle centered at (-(1)/(3), 0) touching w = -1 and w = (1)/(3) on the real axis. (c) Newton-Raphson method derivation for root of f(x) = 0. Step 1: Taylor expand f around x_n. f(x) = f(x_n) + f'(x_n)(x - x_n) + (f''())/(2)(x - x_n)^2. Step 2: Set f(x_n+1) = 0 and neglect higher terms. 0 ≈ f(x_n) + f'(x_n)(x_n+1 - x_n), x_n+1 = x_n - (f(x_n))/(f'(x_n)). Iterate until convergence to six decimals. (c)(i) Newton forward difference for f(2.5). Step 1: Forward difference table (h=1, x_0=2). c|cccccc x & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 31 & 92 & 207 & 394 & 671 & 1056 \\ & 61 & 115 & 187 & 277 & 385 \\ ^2 & 54 & 72 & 90 & 108 \\ ^3 & 18 & 18 & 18 \\ ^4 & 0 & 0 \\ Step 2: u = (2.5 - 2)/(1) = 0.5. Newton forward formula: f(2.5) = f(2) + u + u2 ^2 + u3 ^3. Step 3: Compute terms. u = 0.5 × 61 = 30.5, 0.52 ^2 = (0.5(-0.5))/(2) × 54 = -0.125 × 54 = -6.75, 0.53 ^3 = (0.5(-0.5)(-1.5))/(6) × 18 = 0.0625 × 18 = 1.125. Step 4: Sum. f(2.5) = 31 + 30.5 - 6.75 + 1.125 = 55.875. 55.875

Accepted Answer · 2026-03-18T05:54:19.205Z

(a)(i) Step 1: Recall the formula for z where z = x + iy. z = (x + iy) = x y + i x y. Thus, u(x,y) = x y, v(x,y) = x y. (a)(ii) Step 1: Compute partial derivatives of u. u_x = x y, u_y = x y. Step 2: Compute partial derivatives of v. v_x = - x y, v_y = x y. Step 3: Verify Cauchy-Riemann equations u_x = v_y and u_y = -v_x. u_x = x y = v_y, u_y = x y = - (- x y) = -v_x. Both hold, so f(z) is analytic. (b) Step 1: Let = z^2. For |z| = 1, || = |z|^2 = 1. The map is w = ()/( - 2). Step 2: Solve for in terms of w. = (2w)/(1 - w). Step 3: Impose || = 1. | (2w)/(1 - w) | = 1 2 |w| = |1 - w|. Step 4: Square both sides. Let w = u + iv. 4(u^2 + v^2) = (1 - u)^2 + v^2 = 1 - 2u + u^2 + v^2. Step 5: Simplify. 4u^2 + 4v^2 = 1 - 2u + u^2 + v^2 3u^2 + 3v^2 + 2u - 1 = 0, u^2 + v^2 + (2)/(3)u = (1)/(3). Step 6: Complete the square. (u + (1)/(3))^2 - ((1)/(3))^2 + v^2 = (1)/(3), (u + (1)/(3))^2 + v^2 = (1)/(3) + (1)/(9) = (4)/(9). The image is the circle |w + (1)/(3)| = (2)/(3) (center -(1)/(3), radius (2)/(3)). Sketch: Circle centered at (-(1)/(3), 0) touching w = -1 and w = (1)/(3) on the real axis. (c) Newton-Raphson method derivation for root of f(x) = 0. Step 1: Taylor expand f around x_n. f(x) = f(x_n) + f'(x_n)(x - x_n) + (f''())/(2)(x - x_n)^2. Step 2: Set f(x_n+1) = 0 and neglect higher terms. 0 ≈ f(x_n) + f'(x_n)(x_n+1 - x_n), x_n+1 = x_n - (f(x_n))/(f'(x_n)). Iterate until convergence to six decimals. (c)(i) Newton forward difference for f(2.5). Step 1: Forward difference table (h=1, x_0=2). c|cccccc x & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 31 & 92 & 207 & 394 & 671 & 1056 \\ & 61 & 115 & 187 & 277 & 385 \\ ^2 & 54 & 72 & 90 & 108 \\ ^3 & 18 & 18 & 18 \\ ^4 & 0 & 0 \\ Step 2: u = (2.5 - 2)/(1) = 0.5. Newton forward formula: f(2.5) = f(2) + u + u2 ^2 + u3 ^3. Step 3: Compute terms. u = 0.5 × 61 = 30.5, 0.52 ^2 = (0.5(-0.5))/(2) × 54 = -0.125 × 54 = -6.75, 0.53 ^3 = (0.5(-0.5)(-1.5))/(6) × 18 = 0.0625 × 18 = 1.125. Step 4: Sum. f(2.5) = 31 + 30.5 - 6.75 + 1.125 = 55.875. 55.875

Given that f(z) = SinZ, where Z=x+jy. express f(z) in the form u+jv; (ii) prove that u and v satisfy Cauchy-Riemann equatic A transformation is defined by W = (z+2j)/(z-j) find the image of the circle |Z|=2 in the w-plane and sketch it.

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Given that f(z) = SinZ, where Z=x+jy. express f(z) in the form u+jv; (ii) prove that u and v satisfy Cauchy-Riemann equatic A transformation is defined by W = (z+2j)/(z-j) find the image of the circle |Z|=2 in the w-plane and sketch it.

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