14. Suppose that the complex number z has modulus one, and that 0 < Arg z < 2 TT Prove that 2 Arg(z 1) Arg z

Q: Given 21 = Z3 Prove jk 13 2.

A: Given,  ∠1≅∠3 To prove  j∥k

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

A:

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: 10 2. Use De Moivre's theorem to determine 1 in the form a + ib. 2

A:

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 1: What happens to the above if the root r has a multiplicity m?

A: It is given that, the root r is a root multiplicity m that is fx=x-rmgx where gr≠0

Q: 15) Find The natural domain (a) fx) =3 os X 6 g0) = ©dix) = Sin? x O ra) = 19-x2.

A: given(a)f(x)=3cosxdomain of this function is (-∞,∞)(b)g(x)=13+xx≥0therefore domain is [0,∞)

Q: 2.6. In the text, the modulus is defined in terms of the inner product. Prove at this can be turned…

A:

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

A:

Q: 7. In ABCD, AC and BD divide each other in half at E. (a) Prove that ZDBC ZBDA. (b) Prove that ADBC…

A:

Q: 2.3 Use the Cauchy integral formulae to evaluate 22 - 1 Jo (9 – 2*)(2 + 2i))" where C is a circle |2…

A:

Q: Show that the modulus of the quotient of tow complex conjugate numbers is 1, that is :1.

A: Use the following formulae, to prove the required result. The conjugate of the complex number a+ib…

Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

A:

Q: 4) For A = ajax + azāy + azaz , B = b,ax + b2ay + bzaz, show that A.B cos 0 = A.AVB.B

A:

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: (8.20 ds: Find the tayLor seri's expen sian of In Cxri) about XozO

A:

Q: i) Find an upper bound for the modulus 3z³ + 5z + 1, if |z|<1 where z is the complex number.

A: Now for the mod function: : a+b≤a+b. SO using the same method in the given complex function:…

Q: 3) Use the Second Principle of Finite I nduction to establish that for all nz I o"-1= (a-1) Co to ta…

A:

Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: 1- Find the and argumint g fallowing modulus the Complea numbers 2+i 2+ i 3-i 4i + (Hi)?

A: It is given that

Q: Find all values of z such that cosz = 4.

A: We find all values of z such that cos(z)=4

Q: Theorem 9.If k is even and I, ở are odd positivVe integers, then Eq.(1) has prime period two…

A: Given, if k is even and l, σ are odd positive integers.

Q: Number theory

A: Consider the provided question,

Q: IRI- 2. <| sinz|< -e 2

A:

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: 3=the demmand fumetion forx umits of on item is givem by the functio P= 70-0.07x. what is the…

A:

Q: 1. Prove or disprove that each of the cises 52 of Section 3.1 is cyclic. a. Z2 × Z3

A:

Q: Let A find on Conditions a,bic and d has infinitely that AX =o many solution . what must there Pere…

A:

Q: Theorem 2.8: For positive integers a and b grd co,b) lem ca,b) =ob Coro llory: Icm ca,b) = ob For…

A: Let us consider two pairs of numbers such as 8, 16 and 3,5

Q: 22. Given that Z= 2-3i, express Z/Z* and Z(cos-- isin -) in the form a + bi, where a and b are real…

A: Note: "Since you have asked two questions, we will solve the first question for you. If you want any…

Q: 2.2.6. In the text, the modulus is defined in terms of the inner product. Prove that this can be…

A:

Q: 2. Solve for the complex mumber z in r (4+ 36) – 7 = 7x - 4zi. Show all algebra.

A: Given expression is x(4 + 3i) - 7 = 7x - 4xi

Q: 14. Given: AB | DC; AB = CD; 21 = L2 To prove that DE = BF, what would you prove first? a. AADE =…

A: Given: AB∥DC¯; AB≅DC¯; ∠1≅∠2

Q: The kth harmonic number is defined to be Hk = 1 + 1 2 + 1 3 + · · · + 1 k . Prove that for all n ∈ N…

A:

Q: Let z be a non-zero complex number. D if OsArg(z)sTt/2 D for all complex numbers D if OsArg(z)sn…

A: we know that logzn≠nlogz for all complex numbers. Since ew=z and -π&lt;Imw≤π If w=logz, nw is a…

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: coth- x = In () Q1/ Prove that

A: To prove coth-1x=(1/2)ln[(x+1)/(x-1)]

Q: Show that for A and B of n×n order with complex enteries the matric Ab need not be similar to BA ,…

A:

Q: IRI- 2. <| sinz|< -e 2

A:

Q: 5. Evaluate the integral JI. dA, R = [-2, 4] × [1, 3]. R

A:

Q: By using Caushy Rieman prove that (2) and (3) (2) 4ca) = e (3) .w prove Thot()-

A: (a)

Q: For a Cauchy-Euler DE the roots of auxiliary are m = -3 + 3i then Ye = x= (c, cos(3 In x)+ c2 sin(3…

A:

Q: 4. Prove that an element r of Z2 \ 2Z2 is a perfect square if and only if x = 1 (mod 8).

A: we prove first part⇒let x∈Z2\2Z2 is a perfect square whereZ2\2Z2…

Q: 2. Sketch 0 in standard position if cose = 8. and deter %3D

A: Given that: cosθ=-817

Q: 73. Let Im = 7/2 sin" x dx. (a) Show that Io = 5 and I1 = 1. (b) Prove that, for m > 2, Im : т — 1…

A:

Q: -- a) such that ab =1. Let p>1 be a prime number. Show that for each a + 0 in Zn there is b in Zp…

A:

Q: 12. Prove the following (A+ BB")-B= AB(I+ BT AB)-1 by express the B+BB' AB in two forms. O Type here…

A:

Q: 4- Prove Sca) < ela) and Conclude if ea is la-ea small Compared to lal then: ela) S(a) lal

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.