Let z be a complex number satisfying 12|z|² = 2|z + 2|² + |z² + 1|² + 31. What is the value of z +-? (А) —2 (В) —1 (D) 1 (E) 4

Q: Show that 1 п-1 .. 1 X2 x5 п-1 x2 ... =II("; – 2;) i>j 1 n-1 .. •.

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Q: 1) Prove that there are no positive integer solutions to the equation x2-y?=1

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Q: Prove that |z₁+z₂²= |z₁|²+ z₂², if z₁/z2 is purely imaginary.

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Q: 5. Use De Moivre's Theorem to solve all the roots of x^5 +1= 0 O A. 1 cjs [(nt+2nk)/5] ; k = 0, 1,…

A: Given that x5+1=0. We know that  x+iy=r(cosθ+isinθ(x+iy)n=rn(cosθ+isinθ)n =rn(cosnθ+isinnθ)…

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Q: 4. Consider the complex number w = 3cis (5). For which value(s) of k is x – wa factor of the…

A: I am going to solve the given problem by using some simple geometry to get the required result.

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Q: prove using the IVT. √c +√c + 2 = 3 has a solution.

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Q: Suppose z=a+bi , w=u+vi, a=((|w|+u)/2)^(1/2) , and b=((|w|-u)/2)^(1/2) . Prove z^2=w if v≥ 0 and…

A: Given z=a+bi,w=u+iv,a=w+u212,b=w-u212. Need to show that z2=w if v≥0, and z2=w if v≤0.

Q: Use Fermat's Theorem in your proof that Vn E Z>², 2 is irrational. I

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Q: 5. (a) Describe and sketch the set of points in the complex plane satisfying the inequality: 1< |z…

A: The set of points shown will be bounded by two circles centered at -1,3 of radii 1 and 3.

Q: 6. Show that when ) < |z - 1| < 2, e - ( - 3) (2 - 1)(2 – 3) Complex Analysis n=0 (z-1)" 2+2 2(z-1)

A: We use Laurent series to prove given example.

Q: 4. following is correct? A. B. C. Let p = a + jb, b > 0 be the complex root of the denominator of…

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Q: Prove using the IVT. 2x = bx has a solution if b &gt; 2.

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Q: 2. Let K be a finite simplicial complex and n is equal to the number of path-connected components of…

A: For simplicity, here we use r, instead of n.

Q: (9) Is an complex annullus Ann(2o, R1, R2) with 0 < R1 < R2 simple connected?

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: 1. Motivated by Example 2, let us define a function or trans- formation T from the set of all…

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Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: 3. Find the constants bo, co, a1, b1, C₁ so that So(x)=1+ box + cox² + x³, S(x) = { S₁(x) = a₁ - 2x…

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Q: 6. Show that when 0 < |-1|< 2, (z-1)(z-3) Complex Analysis (z-1)" 2+2 =-36 z−3) ² A=0 2(z-1)

A: We use substitution and Laurent series expansion.

Q: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

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Q: 3. (a) Show that, for any complex number z, zz = |z|², z + z = 2Re(z) and Re(z) < |z]. Hence show…

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Q: %3D n-0o 2n

A: Given. We are given a limit and we need to prove that the limit of the given function at infinity…

Q: 30.Prove that if y1 and y2 have a common point of inflection t0 in I, then they cannot be a…

A: Suppose at the point t0, p and q is zero. That is, pt0=0 and qt0=0. Since t0 is a point of…

Q: Is 1z = 1 for all z ∈ C?

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Q: Let z be a complex number and f(z) = inh( .Then z=0 is a singular point of f but it is not isolated…

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Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: 7. Derive the harmonic conjugate of u(x, y) = x² - y2. (2 2

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Q: 6. Given any two functions f (z)and g (z) and any complex number z0, is it always the case that Res…

A: 6- Yes Assume that f and g are analytic in the neighborhood of z0 By definition of residue, Res(f+g,…

Q: Prove that the equation * + + = 1 has no solutions in positive integers x , Y, Z.

A: x/y + y/z + z/x = 1 if x , y , z are positive integers then case 1 :  If all three numbers are…

Q: 11. If rª +4a12³ +6azx² +4a3x+a4 = 0 is an equation with real coefficients and has two real and two…

A: The roots are concyclic in the Argand diagram means the roots lie on the same circle, or, the…

Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: 2.2.12 Prove that if x and y positive real numbers, then Vx+y # Vx+ Vy. Argue by contradiction. are

A: Let x and y are positive real numbers. Also let x+y=x+y Now squaring on both side,we get…

Q: polynomial in V2+i and compute the conjugates of i over Express i as a Q(/2 + i).

A: As per the question we have to express i as a polynomial in (√2 + i) and then compute its conjugate…

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Q: Find the residue of 3 Z 2 z² - 1 at z→ ∞0.

A: We have to find the solution of the given problem.

Q: Z-1 %3D ,then prove that x2 + y? = (x-1)2+y2 (x+1)2 +y2 (1) If Z %3D Z+1

A: Topic- complex number Note:- as per our guideline if more than one question uploaded than only first…

Q: Prove that |z₁+z₂|²= |z₁|² + z₂², if z₁/z₂ is purely imaginary.

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Q: {1.3.5...(2n - 1)} Prove that: "C, %3D n!

A: You can solve the question from either one of the sides. I solve from right hand side.

Q: Suppose that the complex number z has modulus one, and that 0 < Arg z < . Prove that 2 Arg(z+1) :…

A: Given the complex number z has modulus one, and 0&lt; Arg z &lt; π/2. So, we may take complex number…

Q: 82 핑 3 - 21 Prove that + Cn? m? Jyn=0

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Q: (2.5) Find another representation in which a. r is negative and °<0 < 2n. b. ris positive and -2n <0…

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Q: (2.) Find another representation in which a. r is negative and 0 <0 < 2n. b. ris positive and -2n <0…

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Q: (18) (a) Prove that R* is isomorphic to R>0 x Z2. (b) Is Q* is isomorphic to Qo x Z2? Prove your…

A: a)  To prove :- ℝ*is isomorphic to ℝ&gt;0×ℤ2. Proof :- Define a map ϕ : ℝ&gt;0 ×ℤ2→ℝ* by…