1. Integrate uaong complex variables (step by step) 2. Solve (step by step) 1. Integrate Le...
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(iv) Let a be any nonzero complex number. Show that for 12 – 20/ < |al, Z-Zo 2-20 n=0 n = 0 159)..ila). by a dr =0, 6, 11 d =0 Conclude that Z-ZO Z-ZO a for any closed (piecewise) regular curve y that lies in the disk (z – zo] < |al.
t X and Y be independent random variables with variance ơ1 and ơ3. respectively. Consider the sum. Z=aX + (1-a)% 0 < a < 1 Le Find a that minimizes the variance of Z
linear algebra and complex analysis variables
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1+i 1. Write in standard form x+yi. 2. Find the modulus and principal argument of z = 2 + 2/3 i and use it to show z' = -218 3. Give geometrical description of the set {z:2z-il 4} 4. Find the principal argument Arg(z) when a) z = -2-21 b) z=(V3 – )6 5. Find three cubic root of i. 6. Show that f(z) = |z|2 is differentiable at...
Hi, can you solve the question for me step by step, I will rate
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Question 1 (a) By using implicit differentiation, find dyldx for the curve at the point (0,1). (4 marks) (b) Using the substitution u = sin x, evaluate the integral cos x dx (1- sin x)(2 +sin x) (5 marks) 1+iV3 (c) Evaluate z = in polar form, and hence simplify...
Q1. Solve the complex equation: sinz 3i Q2. Study the analyticity of the complex function fusing Cauchy-Riemann equations: Izl Q3. Evaluate, by using Cauchy's integral formula, the path integral cosh2 z dz (z-1-i(z-4) where C consists of Iz 3 (counterclockwise) Q4. Using the Residue theorem, integrate counterclockwise around the circle C defined by zl 1.5, the following tan z dz Q5. Find, by using parti ial fraction, the Laurent series of the function with center zo 0 for 1< z<3...
#1: Use a change of variables to integrate f (x, y) = y - x over the region described by: –3 <y – 2x < 0 and 0 < 2y – x < 3.
2. Solve the following ODEs using an appropriate method. a) (ex + 1) .y = ev sin x b) dy 1 = -y - dx y=x. x > 0 c) (2x2y3 + 3y2) dy = -xy4 dx Cid
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,
Tutorial Exercise Evaluate the integral using the substitution rule. sin(x) 1/3 1* dx cos(x) Step 1 of 4 To integrate using substitution, choose u to be some function in the integrand whose derivative (or some constant multiple of whose derivative) is a factor of the integrand. Rewriting a quotient as a product can help to identify u and its derivative. 70/3 1." sin(x) dx = L" (cos(x) since) dx cos?(X) Notice that do (cos(x)) = and this derivative is a...
3i)16 in polar form: z r(cos 0isin 0) where (1 Write the complex number z and e= The angle should satisfy 0 0 < 2«.