Given that f(z) = SinZ, where Z=x+jy. express f(z) in the form u+jv; (ii) prove that u and v satisfy…

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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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(a)(i) Step 1: Recall the formula for sin z where z = x + iy. sin z = sin(x + iy) = sin x cosh y + i cos x sinh y. Thus, u(x,y) = sin x cosh y, v(x,y) = cos x sinh y. (a)(ii) Step 1: Compute partial derivatives of u. u_x = cos x cosh y, u_y = sin x sinh y. Step 2: Compute partial derivatives of v. v_x = −sin x sinh y, v_y = cos x cosh y. Step 3: Verify Cauchy-Riemann equations u_x = v_y and u_y = −v_x. u_x = cos x cosh y = v_y, u_y = sin x sinh y = − (−sin x sinh y) = −v_x. Both hold, so f(z) is analytic. (b) Step 1: Let = z². 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(u² + v²) = (1 − u)² + v² = 1 − 2u + u² + v². Step 5: Simplify. 4u² + 4v² = 1 − 2u + u² + v² ⇒ 3u² + 3v² + 2u − 1 = 0, u² + v² + 2/3u = 1/3. Step 6: Complete the square. (u + 1/3)² − (1/3)² + v² = 1/3, (u + 1/3)² + v² = 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(xn+1) = 0 and neglect higher terms. 0 ≈ f(x_n) + f'(x_n)(xn+1 − x_n), xn+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, x₀=2). c|cccccc x & 2 & 3 & 4 & 5 & 6 & 7 f & 31 & 92 & 207 & 394 & 671 & 1056 Δ & 61 & 115 & 187 & 277 & 385 Δ² & 54 & 72 & 90 & 108 Δ³ & 18 & 18 & 18 Δ⁴ & 0 & 0 Step 2: u = (2.5 − 2)/(1) = 0.5. Newton forward formula: f(2.5) = f(2) + u Δ + u2 Δ² + u3 Δ³. Step 3: Compute terms. u Δ = 0.5 × 61 = 30.5, 0.52 Δ² = (0.5(−0.5))/(2) × 54 = −0.125 × 54 = −6.75, 0.53 Δ³ = (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.

Step-by-step solution

Recall the formula for $\sin z$ where $z = x + iy$ .

Compute partial derivatives of $u$ .

Compute partial derivatives of $v$ .

Verify Cauchy-Riemann equations $u_x = v_y$ and $u_y = -v_x$ .

Let $\zeta = z^2$ .

Solve for $\zeta$ in terms of $w$ .

Impose $|\zeta| = 1$ .

Square both sides.

Simplify.

Complete the square.

Taylor expand $f$ around $x_n$ .

Set $f(x_{n+1}) = 0$ and neglect higher terms.

Forward difference table ( $h=1$ , $x_0=2$ ).

$u = \frac{2.5 - 2}{1} = 0.5$ .

Compute terms.

Sum.

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More Mathematics Questions

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.

Step-by-step solution

Recall the formula for sin⁡z\sin zsinz where z=x+iyz = x + iyz=x+iy.

Compute partial derivatives of uuu.

Compute partial derivatives of vvv.

Verify Cauchy-Riemann equations ux=vyu_x = v_yux​=vy​ and uy=−vxu_y = -v_xuy​=−vx​.

Let ζ=z2\zeta = z^2ζ=z2.

Solve for ζ\zetaζ in terms of www.

Impose ∣ζ∣=1|\zeta| = 1∣ζ∣=1.

Square both sides.

Simplify.

Complete the square.

Taylor expand fff around xnx_nxn​.

Set f(xn+1)=0f(x_{n+1}) = 0f(xn+1​)=0 and neglect higher terms.

Forward difference table (h=1h=1h=1, x0=2x_0=2x0​=2).

u=2.5−21=0.5u = \frac{2.5 - 2}{1} = 0.5u=12.5−2​=0.5.

Compute terms.

Sum.

Still stuck on this one?

More Mathematics Questions

Recall the formula for $\sin z$ where $z = x + iy$ .

Compute partial derivatives of $u$ .

Compute partial derivatives of $v$ .

Verify Cauchy-Riemann equations $u_x = v_y$ and $u_y = -v_x$ .

Let $\zeta = z^2$ .

Solve for $\zeta$ in terms of $w$ .

Impose $|\zeta| = 1$ .

Taylor expand $f$ around $x_n$ .

Set $f(x_{n+1}) = 0$ and neglect higher terms.

Forward difference table ( $h=1$ , $x_0=2$ ).

$u = \frac{2.5 - 2}{1} = 0.5$ .