– Using the Residue theorem evaluate dr +1

Q: The number of irreducible monic polynomials of degree 2 over Z5

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Q: z'dz Evaluate where C is z| by residue theorem (z-1)*(z-3)

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Q: Given the characteristic polynomial (r – n) - (r- r2) = 0, find the associated re- currence…

A: The required recurrence relation is, an + (r1r2)an-2 =  (r1+r2)an-1    The detailed solution is as…

Q: Prove that Zo is not isomorphic to Z, X. 10 12

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Q: Abstract Algebra

A: To find the smallest degree polynomial f(x) with coefficients in Q, such that f(√2+√3)=0.

Q: 1-z+2 dr +10+9 Use the Residue theorem to evaluate

A: The integral is I=∫-∞∞x2-x+2x4+10x2+9dx. Consider fz=z2-z+2z4+10z2+9 By theorem, if a > 1 and M…

Q: Calculate the Taylor polynomials T2(x) and T3(x) centered at x = a for f(x) = 28 sin(x), a =…

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Q: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

A: Bijective function

Q: /2- 2

A: We will use basic knowledge of Legendre's polynomials and integral calculus to answer this question.

Q: Evaluate by Residue Theorem/Formula do where a = your registration number first digit, (a + bcos0)²…

A: It is clear that : cosθ=z-1zzdθ=dziz For the given integral ,…

Q: apply Residue theorem to find the 2n cos20 dð. o 3+2sine

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Q: 4) Using the Residue Theorem, show that II 16

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Q: Applying the Cauchy's Residue Theorem, evaluate de (5 – 3 sin 0)²

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Q: ising residue theorem prove dr x+1 2

A: Given:

Q: Find ALL irreducible polynomials of degree 2 or 3 in Zr].

A: An irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the…

Q: Using Cauchy's Residue Theorem, evaluate sin z dz Iz-l<n 2(z – 1n)² 1z-41<π

A: Answer

Q: irr(ꭤ,ℚ) and deg(ꭤ,ℚ

A: Given: 2+i To find:  The irr(ꭤ,ℚ) and deg(ꭤ,ℚ)for the given algebraic number ꭤ ∈ ℂ.

Q: with Zeros of 2i with integer coeRficients that haf E Puno

A: We need to find the polynomial of degree three with integer coefficient

Q: nd to wnwyun g ans 2

A:

Q: (b) Expand and simplify in ascending powers of x as far as the term in x" 1+3x

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Q: (2) Construct the Simpson 3/8-th formula to approximate f(x)dr by using La- grange interpolating…

A: Given: ∫abf(x) dx and the function f(x), where the interval of integration a, b is divided into n=3…

Q: 9. Given the Mac expansion of y = In(1+ x) below, use it to approximate In(0.4) to the 4th decimal…

A: We will use given formula.

Q: Q.No.7.: Using residue theorem, evaluate: dz (2° +4)* dz, c: z-i=2. tion/

A:

Q: Find the linearization at x = a. f(x) = , a = 7 (Express numbers in exact form. Use symbolic…

A: Lx = f(a)+f'(a)(x-a)

Q: Prove that F2 + F = F2n+2 where Fn is the nth Fibonacci number. %3D n+1

A: Given, Fn2+Fn+12=F2n+2 As we know that, F1=1,F2=1,F3=2,F4=2,F5=4,F6=5

Q: 5. Let (1 2 3 A = 2 1 4 1 0 2 Find A-lif possible. If A-does not exist, explain why.

A: Given See the handwritten solution

Q: Proof by induction that 0+0=0+0   where 0 represent x. x+0=0+x

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Q: Coefficient of ' in the expansion of (1+x) (1+ x)* is

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Q: B: Evaluate the integral: TUTO (x + 1)-/2 (x – 1)5/2

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Q: 9. lim xln (1 + (Hint: Indeterminate Product)

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Q: 05- The distinct residue classes comprising F[x]/f) can now be deseribed explicitly; namely, ups…

A: See answer below.

Q: Find the residue f(z) = Z²-22 (2+1)² (2²44)

A:

Q: Write out the complete proof of the integral remainder formula in Taylor's theorem.

A: proof of the integral remainder formula in Taylor's theorem.

Q: Calculate the Taylor polynomials T2(x) and T3(x) centered at x = a for f(x) = 22 sin (x), a =…

A:

Q: Calculate the Taylor polynomials T2(x) and T3(x) centered at x = a for f(x) = 12 sin (x), a =…

A:

Q: x3 + (2i – 1)x³ – 2ix – 5 a = 1- i -

A: Given: The given polynomial is x5+2i-1x3-2xi-5 a=1-i into binomial.

Q: (It v5)" (1 Vsy (b) Prove that f. for all integers n> 1.

A:

Q: c) Prove that there cannot exist two different polynomials q,r E K[x], both of degree less than n,…

A:

Q: 1. Use the definition to find the first five nonzero terms of the Taylor polynomial, centered at c =…

A: To find the Taylor polynomial of function f(x) =1/x centered at c=1

Q: x +1 x3 +1

A:

Q: Show that V3 € Q[6]

A: Q is the field of rational numbers such that…

Q: 3) Let the order on the X÷1-00₂ √₂)NQ 2 be the known order. find the Supremum Supremum and infimum…

A: Here given set is A=-2,2∩ℚ

Q: (*;)-()0)

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Q: Find the minimal polynomial satisfied by √2 + √5. over Q.

A: We have to find the minimal polynomial which satisfied by 2+5 over Q.

Q: Find the field of quotients of integeral domain z[i] and z[√2]

A: We will solve this question using basics of ring theory and just the definition of Fields and…

Q: Show that x4 – 22 x2 +1 is irreducible over ℚ. [Hint: Show that it has no factor of degree 1 and…

A: Given the given expression is x4-22x2+1. Use theorem 23.11 and corollary 23.12.

Q: Find the residue of f(z)=- at z=1 (z-1)* (z-2)(z-3)

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Q: 6. Find the third degree Maclaurin polynomial for f(x) = cot-(x) and use this to approximate…

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Q: Find the Taylor polynomials of orders 0, 1, and 2 generated by f at a. (question 8, part 1 in image)

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Q: state and prove the Cauchy Residue theorem?

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