4) Let C be a circle with center at a e C and radius R > 0. For any complex number z, let z* denote its symmetric point with respect to C. Prove Ptolemy's theorem using the fact that for any two R2 complex numbers z1 and z2, neither being a, we have |2 – 2| : |21 – z2|. |21 – a| |22 – a| -

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

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

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

A: See solution below

Q: The complex numbers 0, z+ 1 and z2 are distinct and collinear on the complex plane. Draw the locus…

A: As per our guideline we are supposed to answer only first question of a multiple type questions.…

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

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Q: If z1 and z2 are complex numbers such that |21| = |2|, prove that arg(2122) = arg ((21 + z2)²) .

A: This  problem is related with complex analysis. Given that z1 and z2 are two complex numbers such…

Q: We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn)…

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Q: Prove that there is no complex number z such that 1 |z|«anda,z' = 1, where|a,| < 2. 3 r=1

A: In this question we have to prove that there is no complex number z such that z&lt;13 and ∑r=1n…

Q: The set of all complex numbers is denoted by C={x+iy:x,y ER} 0 صواب İhi O السؤال 2 lim does not…

A: Complex numbers: A complex number is represented in the form a+ιb, where a and b are real numbers,…

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: If Zo is any complex number interior to any simple closed contour prove that dz 2πί, η- 1 Jc (z-…

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Q: Let f(2) = Ln(2² + 1). Find a possible arrangement of the branch cut of f(2) on the complex plane.

A: Given: f(z)=Ln (z2+1)

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: Every complex number a has a 4-bth root in C That is, there exists a complex Mumber w such that…

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Q: 1- The distance between two complex numbers z1 Z2 is denoted by: a) Iz – z2| b) /x + y c) Vz, + z2…

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Q: Sketch the set S of points in the complex plane satisfying the given inequality. Determine whether…

A: To find- Sketch the set S of points in the complex plane satisfying the given inequality. Determine…

Q: A set S is said to be convex if each pair of points P and Q in S can be joined by a line segment PQ…

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Q: P,Q,R,S are four non-collinear points in the Argand diagram, representing the complex numbers…

A: Given that z1, z2, z3, z4 are four non-collinear points in Argand plane. Consider a quadrilateral…

Q: For complex number z and w, prove that zw-wz=z - w if and only if z =w or, zw = 1.

A: We know that , for a complex number|z|2=zz¯and when z is a real numberz=z¯

Q: Given that z = cos + isin and u-iv-(1+z)(1-22). Show that 30 v = utan 2 r = 4² cos² (2). where r is…

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Q: Let z2T3-L p=S+Si O fird all the complex mumbers %3D guien by hza

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Q: V3 1 (- 2

A: given(-32-12i)20 =(-1)20(32+12i)20 =(cos30+isin30)20 The formula of Demoivre's theorem is cos θ+isin…

Q: 13. Prove that the three points (-3, 2, 4), (6, 1, 2), and (-12, 3, 6) are collinear by using the…

A: To solve the following problem

Q: 3 If C is a circle of radius r with centre z, in the complex z-plane and if n is a non-zero integer…

A: Consider the given integral as, ∮dzz-z0n+1

Q: Find the centre and radius of the circle formed by all the points represented by z = x+ iy…

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

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Q: Given that z is a complex number, sketch the common region shared by the following locus: |z – 2| 6

A: let z=x+iy

Q: Use Bombelli’s approach to find a complex number c = a + bi such that c3= 18 + 26i where a and b are…

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Q: If Zo is any complex number interior to any simple closed contour C prove that dz 2πί, η -1 c…

A: Let C be a simple closed contour and let z0 be a complex number inside C. Prove that:…

Q: Write down the definition of the complex conjugate, z, if z = x + iy where x, y are real numbers.…

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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: Let z be a complex number. Then sin(3z)-i is bounded in C. False True

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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: The points A, B, C and O represent the numbers z, =, 1 and 0 respectively in the complex plane.…

A: We will treat A,B,C as position vectors to do the proof

Q: 33. , Given that the complex numbers z - 3 + i and z2 = -3 - i are such that Iz - z,|-|z – z2 l. The…

A: Here we find locus of z

Q: ) The complex conjugate of z is denoted as z. Solve for z in the equation: 27(1 – j2) 2 - j3 5z 1+j2…

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Q: Which of the following represents the set D on the complex plane?

A: Given set: D=z∈ℂ:z-1+i&lt;2. We have to find the graph which represents D.

Q: What are the perpendicular generators on (P) to 4 %3D of the surface

A: Given equation s are, x216-y24=z....1x3=y2=z-4...2 Equation of generators of the hyperboloid (1) are…

Q: Answer the Following: 1. describe the set of points z in the complex plane that satisfy (a) =:-1 (b)…

A: Complex representation in argand plane

Q: For all complex numbers Z1,Z, prove that |2z|+|*z|s]°z+z[

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

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Q: 15. On the Argand plane, carefully sketch the set of complex numbers, z, that satisfy the condi-…

A: Consider the given condition, z=argz Let's consider that the argument is principal argument and…

Q: Z Is any complex number interior to any simple closed contour dz 2 Ti, n= 1 prove that for all…

A: It can be solved using cauchy integral formula

Q: Let z be a complex number such that Iz| = 4 and arg(z) = 5Tt/6, %3D * = then z

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