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Complex Analysis

Complex Variables Convergence of Summations

A) Prove that sum(z^n/n) converges at every point of the unit circle except z=1 although this power series has R=1. b) Use partial fractions to determine the following closed expression for c_n c_n=((1+sqrt5/2)^n+1 - (1-sqrt5/2)^n+1)/sqrt5 Ps. Here c_n are Fibonacci numbers defined by c_0=1, c_1=1,.... c_n=c_n-1 + c_


Suppose that f is holomorphic in a region G(i.e. an open connected set). How can I prove that in any of the following cases a)R(f) is constant b)I(f) is constant c)|f| is constant d) arg(f) is constant we can conclude that f is constant. Ps. here R(f) and I(f) are the real and imaginary parts of f.

Catenary Model Heights

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

Logarithm and Complex Numbers

See the attached file. 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:

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 a

In the real world, where might these so-called imaginary numbers be used?

My question which I am not exactly sure is : 1.I need to find the mass and radius of three of the nine planets in our solar system. I need to be sure that the masses are expressed in kilograms and the radii are expressed in meters. 2.Using data, I need to find out how to calculate the gravitational acceleratio

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.

operation of Complex Numbers 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


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.)

Integration, Primitive and Fundamental Theorem of Calculus

Compute the integral in each of the following three ways: ? Directly, by integrating the function along C. ? By finding a primitive and using fundamental theorem of calculus ? By integrating the same function along the straight line from -1-i to 1+I and using Cauchy's closed curve theorem. Please see the attached file