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# Complex Analysis

### Complex Variables: Rectifiable Path

Please help with the following problem. Fix w=re^i(theta)(not equal)0 and let gamma be a rectafiable path C-{0} from 1 to w. Show there is an integer k such that (integral)gamma z^-1 dz = log r + (theta)+ 2 pi i k See attached file for full problem description.

### Mathematics: Solving Complex Equations

Please see the attached file for the fully formatted problems.

### Finding the real and complex roots of a quadratic equation.

2x2 - 3x + 6 = 0

### Solving Complex Variable Equations : DeMoivre's Theorem

Find the values of: i^(Sqrt(3)) The answer is: cos(pi*sqrt(3)[1/2+2k]) + i.sin(pi*sqrt(3)[1/2+2k]), for any integer k. keywords: de moivres, de moivre's

### Solving Complex Equations

I need to solve for all of the roots of (z+1)^4 = (1-i). Any idea on how to do it? keywords: imaginary

### Solving Equations : Real and Complex Solutions

Find all real or imaginary solutions: (attached). 6x = -(19x+25)/(x+1)

### Lim x--->2 sqrt x^2 + 5

Please explain how/why this is done: lim x--->2 sqrt x^2 + 5

### Verifying a Complex Root

Verify the z = (3-2j) is a root of the polynomial 2z^4 - 18z^3 +66z^2 - 102z + 52 and hence find the other three roots.

### Simplifying Complex Expressions

Write this expression in the form a + bi, where a and b are real numbers: (2 - 3i)(2 + 3i)

### Solving Systems of Equations with Complex Numbers ( Cramer's Rule )

Showing all working, solve the following pair of simultaneous equations for i1 and i2, expressing the answers to exact whole numbers: (3 - j4)i1 + (8 - j7)i2 = 40 - j38 (2 + j3)i1 - (5 + j2)i2 = -6 - j8

### Fixed and Variable Cost Analysis

Costs can be classified into two categories, fixed and variable costs. These costs behave differently based on the level of sales volumes. Suppose we are running a restaurant and have identified certain costs along with the number of annual units sold of 1000. Item: Raw Materials (cost for hamburgers) Total Annual Cost: 650

### Rectilinear motion problems

See attached file for full problem description. 9. The accompanying figure shows the velocity versus time graph for a test run on a classic Grand Prix GTP. Using this graph, estimate a) The acceleration at 60 m/h (in units of ft/s^2) b) The time at which the maximum acceleration occurs. 19. The position function of a

### Lambda Equation

Let lambda be real and lambda > 1, Show that the equation ze^lambda&#8722;z = 1 has exactly one solution in the disc |z| = 1, which is real and positive.

### Networks and Project Analysis : Critical Path, Precedence Diagrams Start and Finish Dates

The marketing department at Bodnar Industries is developing a promotional campaign to introduce a new product. A listing of the various activities required, their immediate predecessors, and estimates of their times (in days) is given below. a) Draw the precedence diagram for this network. b) Find the means and standard d

### Complex Numbers, Standard Form and De Moivre's Theorem

1. Evaluate the expression and write your answer in the form a+bi: i^100 2. Prove the following properties of complex numbers. 3. Find all solutions of equation:- 4. Find the indicated power using De Moivre's Theorem.

### Solving Differential Equations Using Laplace Transforms

Find Y(t) for all t Y''+2Y'+2Y = h(t) Y(0) = 0 Y'(0) = 1 = {0 (less than or equal to) t ( less than)Pi and t ( greater than or equal to) 2 Pi h(t) = { 1 Pi (less than or equal to) t (less than)

### Find all cube roots of -8, in rectangular coordinates

Find all cube roots of the number -8 and state the final answer in rectangular coordinates.

### 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

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

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). 2. Whe

### 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_

### Holomorphisms

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

### Expanding with Complex Numbers

Perform the following complex number multiplication and write the answer in standard form: (-3+3i)(2-i)

### Riemann Sums Formatted Problems

Problem: Evaluate by interpreting it as the limit of Riemann sums for a continuous function f defined on [0, 1]. My work: = =

### 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 the Locus of All Points in the Complex Plane

Please give details of soln |z-i| + |z| = 9.

### Solving for the Locus of Points in the Complex Plane

|z|= |z-i| z = x + yi.

### Finding real or imaginary solutions

Sqrt 7x + 29 = x + 3 keywords: complex

### Find all real or imaginary solutions to the equation

Find all real or imaginary solutions to the equation. Use the method of your choice. 3v^2 + 4v - 1=0

### Simplifying Complex Numbers

Simplify the complex number i^59 as much as possible