Let a and z be complex numbers with |a| # 1 and |2| = 1. %3D z - a āz – 1 Show that w= satisfies |w| = 1.

Q: Q: For any two complex numbers z, and zz, prove that z-z|+|z+!z|=z-z-2+z-z+

A:

Q: Let z be a complex number. Then cos(3z)-i is unbounded in C. * False True

A:

Q: Let z be a complex number. Then cos(5z)+i is bounded in C.* O False True

A:

Q: Let a, b, and c be any three nonzero complex numbers. If |z| = 1 and 'z' satisfies the equation az +…

A:

Q: Let , ² be the complex cube roots of unity. Let z₁, Z₂, Z3 be complex numbers such that Z₁ + Z₂ +Z₁…

A: Given that, Let ω,ω2 be the complex cube roots of unity. Let z1,z2,z3 be complex numbers such that…

Q: Derive the canonical form of the given parabolic PDE (b) - 3u. +64.- 3u = cos(x).

A: given 2nd order PDE -3uxx+6uxv-3uyy=cosx   ...i claim- to derive the canonical form of given PDE

Q: A if z, and z, are complex numbers Z2 Show that ()= Z 22

A:

Q: 5. Determine the roots of the complex numbers studied in class. z³ = 2 – 3i

A:

Q: How may the following complex numbers be obtained fromz = x + iy geometrically? Sketch. 1 / z

A:

Q: Draw f (E) in complex numbers system if f (z) = z +Z + izz and E = {z e C:1 < |z| < 3}. %3D

A: Given is a function f(z) and set E. First we let z = x+iy where x and y are real numbers. Then…

Q: Let z be a complex number such that |z|=1, that is z∙z ̅=1. Compute 〖|1+z|〗^2+〖|1-z|〗^2.

A: Given that -  |z| = 1 we have to compute the following -  1+z2 + 1-z2

Q: Find the largest subset of the complex plane on which f(z) = cos(z)e-² is holomorphic, and the…

A: Given :  f(z) = cos(z¯) e-z¯ To find : We have to find the largest subset of complex plane on which…

Q: We consider the Complex numbers x,y, z with modulus 1 with the property that: x + y + z = 0 and the…

A: Trigonometry is a branch of mathematics that studies relationships between side lengths and angles…

Q: The modulus of the complex number z = 1- e" is A 2sin (0) B. 2 cos (e) В. C. 2 D. 1

A: We have to find the modulus of the complex number z=1-eiθ.

Q: 1. Determine all the complex numbers for which f(z) = Log(z – 1+ 2i) is differentiable. |

A: Since you have posted multiple questions. I am answering the first one. Please post the question…

Q: The set of all complex numbers is denoted by C={x+iy:x,yER} ylgn O ihi O السؤال 2 lim does not exist…

A: Every complex number is x + y i , where x and y are real number So statement is True.

Q: If the complex number z = x + iy satisfies the condition |3z – 6i| = 3 , then z lies on Oa. a.…

A:

Q: Let z be a complex number. Then cos(32)-i is unbounded in C. * False True

A: use:- cos(x+iy)=cosxcoshy−isinxsinhy

Q: Let z be a complex number. Then Im(-iz+1)= * O =liz-1| O None of these Ozliz-1| Sliz-1|

A: The last option is correct.

Q: 1) If we have the following complex numbers: Z, =2+3i, Z, =3-2i %3D Z, (d) Find (a) Z, + Z2 (b) Z,…

A:

Q: D dz |z|=1 |z-a|² 2

A:

Q: 3. If z, and zz are two complex numbers, prove that

A: Let's find.

Q: Let z be a complex number. Then Im(-iz+1) = * O =liz-1| None of these O zliz-1| O sliz-1| OOO

A: Let z be complex number then

Q: 5. Determine the roots of the complex numbers studied in class. z3 = 2 – 3i %3|

A:

Q: Find the modulus (r) for the given complex number z = -6.3 + 8.2i.

A:

Q: Let z be a complex number. Then sinh(2z)-i is bounded in C.* False True

A: Given a statement

Q: 1. Consider the nonzero complex numbers z, = 2i, zz = 8i and z; = 7i. (a) Evaluate (z,z,z,) (2) +2,…

A: Introduction: In most cases, the primary value Arg(z) of a complex number z=x+iy is equal…

Q: 1. If z = 1+ iy is a complex number. Use the exponential form to compute (1+ /3 i)201

A: Definition used -            DeMoivre's theorem -             For a complex number z=r(cosθ+isinθ)…

Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=1/e is (are) * None of these does not…

A:

Q: 4. This question has two parts. a) Compute the value of eln 2020+3xi/4 b) Find all complex numbers z…

A:

Q: 3. First, show that if we define the complex number z = a + bi to be the matrix and a a -b -d w = c…

A: given z=a-bba  and w=c-ddc multiply the two complex numbers  zw=a-bbac-ddc=ac-bd-ad-bcad+bcac-bd…

Q: Let a be a complex number with 0 < |a| < 1. Then the set |z – al < |1 – āz| is: (a) a square with…

A:

Q: The modulus of a complex number x+jy is equal to

A: The modulus of a complex number x+jy is equal to

Q: 4. Let z = -4 + 3i and zw = -7 – 2i. By showing clear steps, find w in the form of %3D a+bi.

A:

Q: Let z be a complex number. Then Im(-iz+1)= * O =liz-1| None of these O 2liz-1| <liz-1|

A:

Q: Let z be a complex number satisfying 12|z|² = 2|z + 2|² + |z² + 1|² + 31. What is the value of z +-?…

A: Given that Let z be a complex number satisfying  12z2=2z+22+z2+12+31 To find the value of z+6z

Q: Let z be a complex number. Then cos(72)+3i is unbounded in C. * O False True

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: Let z be a complex number. Then cos(72)+3i is unbounded in C.

A: A complex no. in the form of a+ ib

Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=e is (are) * O z=1 None of these does…

A:

Q: The complex number modulus z = 12 - 5i is:

A: Given,         The complex number is z = 12 - 5i .

Q: 2. (а; Sketch each of the following sets of complex numbers z that satisfy the given…

A: We will use basic complex analysis

Q: Let , ² be the complex cube roots of unity. Let z₁, z₂, z, be complex numbers such that z₁+z₂+z3 = A…

A: We will use properties of cube root of unity  ω3=1  and  1 + ω + ω2 =0

Q: The complex numbers an, n > 0, are defined by the condition that Σan (z − 2i)” = 1+2z + 3z² + · · ·…

A:

Q: Suppose a complex number z satisfies the equation (1+ 2z)³ = eiª (1 – 22)³, for some a E (T, 27).…

A: Given equation is 1+2z3=eiα1-2z3 where α∈π,2π

Q: Question 9: Given that z = cos + i sin and u- iv = (1 + z)(1 − j²z²). Show that V = utan 30 2 r = 4²…

A:

Q: Let z be a complex number. Then cos(tz)+3i is unbounded in C. * False True

A:

Q: 2(2) A complex number z = v2(cos+i sin ), find In ( Z.Z' )

A: z = √2(cosπ/4 + i sinπ/4)  z = √2(1/√2 +i/√2 )  z = (1+i)  z* = 1-i