Explore BrainMass

# Complex Analysis

### Complex Numbers : Expand and write in Standard Form

(-6 -4i)(1 - 5i)

### 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

### Complex Numbers : Standard Form

Perform the following and write answer in standard form ( 2 + 4i) (1 - 2i)

### Simplify the expression

(3- 2 principle square root 7) (3+2 principle square root 7)

### 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:?

### To represent the complex number is polar form.

To represent the complex number is polar form. See attached file for full problem description.

### 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

### Solving for Complex Numbers: Standard Form

Given this expression: (6 - 3i) - (-2 + 7i) How can you perform these operations and then use standard form to write the result.

### Complex Number Singularities

Singularites (See attached file for full problem description)

### Calculus

Question 1 Multiple Choice The two sides of a right triangle have lengths 2.92 and 3.98. Find the hypotenuse. &#9633; 6.90 &#9633; 3.34 &#9633; 4.94 &#9633; 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 Number Operations and Polar Form

All decimal answers need 4 place values beyond the decimal. 2. (-3 + 2i)² 3. 2i * 3 convert the following to polar form: *radian mode 4. -12 5. 6-3i.

### 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: Radius of Convergence

Suppose ... has radius of convergence R. Find the radius of convergence of: ? ? Please see the attached file for the fully formatted problems.

### 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

### 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

### Complex Variables : Use definition of limit to prove .... Finding Epsilon and Delta

Use definition of limit to prove .... Please see the attached file for the fully formatted problems.

### Complex Analysis and Function

Differentiability (See attached file for full problem description) --- Find all functions f=u+iv which are differentiable everywhere and which have: ? u(x,y)=y ? u(x,y)= ---

### Complex Analysis for Holomorphic Functions

Holomorphic. (See attached file for full problem description) --- 1) Let . ? Show that f is holomorphic in . ? Find its derivative f' and show that f' is holomorphic in . 2) Define a function f by where . Determine all points z where f is differentiable. ---

### Complex Variables : Stereographic Projections, Hyperbolic Functions and Integrals and Circles

Please see the attached file for the fully formatted problems. 1. Let P1 and P2 be two points on the unit sphere x2 + y2 + z2 = 1, and w and w2 the corresponding points on the plane z = 0 under stereographic projection. Show that if P1 and P2 are antipodal points on the sphere, then W1W2 = ?1. 2. The hyperbolic functions sin

### Complex and Real Solutions

Let A be a complex number and B a real number. Show that the equation |z|^2+Re(Az)+B=0 has a solution if and only if |A|^2 >= 4B. If this is so, show that the solutions set is a circle or a single point.

### Plot complex numbers on an Argand diagram

(See attached file for full problem description)

### Conversion of Complex Numbers to Cartesian and Polar Form

Convert each of the following to polar form. 1. 9 - j5 giving the argument in Radians 2. 9 + j16 giving the argument in Degrees Convert each of the following to polar form 2. pi / 7 Please see the attached file for the fully formatted problems.

### Complex Expressions: Cartesian and Polar Form

For questions 1 - 3 give answer in Cartesian form and question 4 in polar form. Q1 2z1 + z2 - 4z3 where z1 = 5 - j7, z2 = 4 + j, z3 = 8 - j5 Q2 z1z2 where z1 = - 3 - j5, z2 = 4 + j5 Q3 where z1 = 1 + j6, z2 = 3 + j7 Q4 Please see the attached file for the fully formatted problems.

### Solve the Complex Rational Expression :

(1/x + 1/y)/(x+y) = See attached file for exact problem.

### Calculus Critical Point Derivatives

1. Given f(x,y) = x^2-4xy+y^3+4y Find the critical points and then use the Saddle Point Derivative Test to determine if they are max, min, or saddle points. 2. Given f(x,y)=4xy-x^4-y^4 Find the critical points and then use the Saddle Point Derivative Test to determine if they are max, min or saddle points. 3. Find the