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Complex Analysis

Stereographic Projection

1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1). a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1 ps. here z bar means conjugate of z b) Show that the triangles Nz'z and NZZ' are similar. The

Projective geometry hyperplane

Projective Geometry Problem 4 Let C be the curve in a complex affine plane E. Find the infinite points of C, i.e. the points of the projective closure of that lie on the hyperplane at infinity. See attached file for full problem description.

Catenary Model

A) Graph the model b) Find the heights of the cable at the towers and at the midpoint between the towers, and c) Find the slope of the model at the point where the cable meets the right-hand tower y = 18 + 25cosh x/25, -25 ≤ x ≤ 25

Riemann Sums

Please see the attached file for the fully formatted problems.

Logarithms and Complex Numbers

1. Given that s = 1.59t(1-3v), obtain the value of v when s = 3.52 and t = 21.56. 2. Solve log(2x + 3) = log(4x) + 2, for x giving the answer correct to 3 significant figures. 3. For a thermodynamic process involving a perfect gas, the initial and final temperatures are related by: T1 exp(^s/Cp) =

Finding a Vertex and Vertex Form and Dividing Complex Numbers

1.Find the vertex form of the quadratic function g(T)=2T^2-4T+5 and determine the coordinates of this functions vertex vertex form_______________ the vertex______________ 2. solve problem following equations algebraically showing all work and steps and solutions -2X^4+6X^2-4=0 X=__________________ 3.showing all

Argand Diagram and Complex Impedance

Please see the attached file for full problem description. Show the following complex numbers on an Argand diagram: Given that the equivalent impedance of parallel complex impedances in an electric circuit is given by: calculate the equivalent impedance br the following circuit.

Question: complex number in polar form

(See attached file for full problem description) 1. simplify each of the following: giving your answer as a complex number in polar form: 2. Convert the following complex numbers in polar form to rectangular (a+jb) from:

Algebra: Complex Roots and Solutions, De Moivre's Theorem and Argand Diagrams

1 Find the real and complex solutions of these cubic equations. a) (z-3)(z2-5z+8)=0 b) z3 - 10z2- 34z- 40 = 0, given that 3-i is a root (solution). 2 Solve the equation z3 = 125 cis 45 3 Consider the complex number: z = = cos + z sin a) Use De Moivre's theorem to find z2, z4 and z6. Leave your answers in polar form. b) Pl

Complex numbers

Singularites (See attached file for full problem description)


Indicate which answer is a simplification. (See attached file for full problem description)


Question 1 Multiple Choice The two sides of a right triangle have lengths 2.92 and 3.98. Find the hypotenuse. □ 6.90 □ 3.34 □ 4.94 □ 3.20 Question 2 Multiple Choice An equation used in the study of protein molecule is In A+ In h - In(1 - h) Solve for

Complex Analysis : Homotopic and Holomorphic Functions

Let be curves in an n open set U. suppose is homotopic to and is homotopic to. Show that is homotopic to . Let where h(z) is holomorphic on an open set U, and h(z) for . Let C be a closed curve homologous to 0 in U such that . Show that Please see the attached file for the fully formatted problems.

Complex Numbers Operations and DeMoivre's Theorem

Apply DeMoivre's Theorem to solve the following: *any angles that are multiples of pi/6, pi/4, pi/3, pi/2 or pi must be converted back into regular complex form! 6. (-1-i)^4 7. (-4sqrt3 + 4i)^1/3 8. (2(cos pie/3 + i sine pie/3))^5 9. z^4 +8sqrt3-8i=0; solve for z

Complex Numbers : Polar Form and DeMoivre's Theorem

6. Please explain step by step Apply DeMoivre's Theorem to find (-1+i)^6 * change to polar form first You will recognize the angle, so put in the correct value of sine and cosine to reduce back to simple complex form 7. Find the fourth roots of 16(cos pi/4 + i sine pi/4) ; n=4 Please explain in detail 9 solve for

Complex Analysis : Extended Liouville Theorem

If f is entire and if, for some integer there exists positive constants A and B such that for all z, then f is a polynomial of degree at most k. (Hint, use the function Prove that and use induction on k.)

Complex Analysis : Primitive F and Cauchy-Riemann Equations

(See attached file for full problem description with symbols and equations) --- Let for . Prove in the following two ways that f has no primitive: a) Assume that f has a primitive F (i.e. these is an entire function F with F'(z)=f(z) for all z). Show that f then would have to satisfy the Cauchy-Riemann equations. Check t