Find the Laurent expansions of the following around z = = 0: (i) (z – 3)-1, valid for |2| > 3; (ii) 1/(2(1 – z)), valid for 0 < ]z| < 1; (iii) z³el/z, valid for |2| > 0; (iv) cos(1/2), valid for |2| > 0. -

Q: Let = e²i/n be an n-th root of unity. Show that (k(1-n) Resck (27²1) n for all 0 <k<n-1. =

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Q: Newton's method is concerned with the following recursion formula: f (xn) f'(xn)' Xn+1 = Xn…

A: Given that Newton's  method is concerned with the following recursion formula:…

Q: (a) 5 Find the coefficient of x in the expansion of (2x -) . (b) Hence find the coefficient of x in…

A: You can use the binomial expansion to solve the problems as follows :

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A: To find the Taylor polynomial for given function.

Q: JUX² +2x-8 dx Evaluate lx + 1Wx + 1)* + 9 = l(x + 1) + Vtx + 1}² – 9|]+c 9ln (+ 1)/(x + 1)² – 9 –…

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Q: If the first five terms of the Taylor expansion for f(x) about x = 0 are 2-3x + === x² + 7 then…

A: we are given the Taylor series expansion for a function f(x) and we need to identify f'''(0)

Q: Given: sinx = (-1)" (2п+1)! x2п+1for x € R n=0 Compute the exact value of the sum 0-1)"(2n+1) „2n+…

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Q: 6n2 + 3n = 3 2n2 – 5 Prove: lim

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Q: (d) (27+5)(2- i)|=/3 12z +5|. %3D

A: Solution D: We have to prove (2z¯+5)(2-i)=3(2z+5) L.H.S. Let z=x+iy So z¯=x-iy therefore…

Q: . Given P(x) =a( b – x) ( x+ c) ( dx+ 1) , where a, b,c and d are Re Z*. If the first term of the…

A: Given P(x)=a(b-x)(x+c)(dx+1) 1st term of expansion=-8x3 P(-3)=0

Q: 1. 2. 3. If the first five terms of the Taylor expansion for f(x) about x = 0 are 5 3 3-7x+ x² +…

A: The given problem is to find the value of f'''(0) and first three terms of taylor series expansion…

Q: 7. Evaluate * (D5 - 3D4 + 8D³ + 16D² - 9D - 13)y = 0 y = (C₁ + C₂x)e¯x + С3ex +C4e²x cos3x + C5e²x…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: 3. Let f(n) be defined for n >0 be defined recursively by f(0) = 7, f(1) = -5, f(n + 2) = (f(n) –…

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Q: röve the following identity () - (":")- (:-) n n n | k k k 1

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Q: Give combinatorial proofs of the following identities. (a) {"} = 2"-1 – 1. (b) {„",} = (3). (c)…

A: Given: Let us show the combinatorial proofs of the following identities a) n2=2n-1-1 b) nn-1=n2 c)…

Q: 2p 4-1(e) + Joi(e) = 空p(e). %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: ;0<x<0, - 0 <0 <0 Let X,,X,X,r(x;e)=- e.w Discuss the sufficiency by using Factorization Theorem.

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Q: None

A: Given, a)   f(x)=(x-2)2  b)   f(x)=(x-1)(x+1)

Q: How large should n be to guarantee that the Trapezoidal Rule approximation to [(-2¹-…

A: and

Q: numerically greatest

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Q: all odd degree terms in the expansion of (x + v(x³-1)5+ (x – v(x³-1)5, (x > 1) is

A: Introduction :

Q: 1.) (2a* –be)(ša² +3bc) (x*y-5z')(x*y–42*) 2.)

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Q: The principal at x=o of f(x) = √₂² +2² - x - x² (with respect to up(x) = x) is: (c) x8/7-x² (d)=x²…

A: To find the principal part of the given function  as with respect to g(x)=x, we will use the…

Q: Prove the following identity: (x + y)2- (x-y)? = 4xy %3D

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Q: Q.4) Find the value of Sigma (d=sum of student id + 22, and c=12) 4c(d + x)(7c* +2cz +9z*) ((d +x)*…

A: As the student ID is not provided, I assume the sum of the digits are 10 (say). Then d=10+22=32 and…

Q: Show that the generating function for compositions into an even number of parts, each of which is…

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Q: f(z)=z²-(6+-4i) z+(-27+12) has 2 roots: z₁ and z2, sorted in an increasing manner according to the…

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Q: Let f(2)= (z+4)(z − 2)³* (6.1) Calculate the Laurent expansion of f about the point z = 2 in the…

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Q: 5. In this problem you will need to use complex number arithmetic. Assume that the equation f(x) = 0…

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Q: (a) (b) (c) 6. The Laurent expansion of f(z) (d) (e) 1 in the domain defined by [2] > 3 is az. z-3…

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Q: cos z Find the Laurent expansion for /(2) = expanded in powers of (z+). (4) (a) Find the residue of…

A: Given function is fz =  coszz+π26

Q: (1– (C+D)), provided (C+D) 4PQ. (28) Substituting (25) and (26) into (28), we get the condition…

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Q: Find the first three terms of the sequence (B₁f) of Bernstein polynomials for f(x) = x³.

A: The sequence of Bernstein polynomials for a function f(x) is defined as follows:B_n^k(x) = C(n, k) *…