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Complex Analysis

Dividing by Zero and Imaginary Numbers

Mathematicians say that division by zero is forbidden. The expression 5/0, for example, is undefined. "Undefined" in this sense means "unable to be determined". Why is this? When we divide 5 by 5 (5/5) we get 1. Divide 5 by 2 we get 2.5. Each time we make the denominator smaller in the expression 5/x, the expression gets

Determine if the following equations are real or complex.

Determine if the following equations are real or complex; explain the answer in detail. Determine whether the following equations real or complex solutions; justify your answer. Note: It is not necessary to find the solutions; just determine if they are real or complex and explain why. a) 5x2 + 8x + 7 = 0 b) (7)1/2y2 - 6

Are the roots real, repeated real, or complex?

Determine if the following have a solution or not? justify answer. (apply the discriminant) are the roots real, repeated real, or complex? 1) 5x^2+8x+7=0 2) (7)^1/2y^2-6y-13(7)^1/2=0 3) 2x^2=x-1=0 4) 4/3x^2-2x+3/4=0 5) 2x^2+5x+5=0 6) p^2-4p+4=0 7) m^2=m+1=0 8) 3z^2+z-1=0

Solve the triangle and complex number

Please see attached file for full problem description. 1. B = 54 degrees, C = 112, and b = 18 2. Solve the equation on the interval [0, 2pi]: (cosx)^2 + 2 cos x + 1 = 0 3. Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. C = 110°, a = 5, b = 11 4. Solve the triangle.

Quantitative Analysis - Waiting lines and queuing Theory models

I have some Quantitative Analysis questions I need help understanding. Waiting lines and queuing Theory models 1. The New Providence shopping mall is considering setting up an information desk manned by one employee. Because of the complex design of the mall, it is expected that people will arrive at the desk at about twi

Complex Variables, Laurent Series and Uniform Convergence

(1) Let G = {z : 0 < abs(z) < R} for some R > 0 and let f be analytic on the punctured disk G with Laurent Series f(z) = sum a_n*z^n (from n = -oo to oo). (a) If f_n(z) = sum a_k*z^k (from k =-oo to n), then prove that f_n converges pointwise f in C(G,C) (all continuous functions from G to C (complex)); i.e., {f_n}

Derivation of Poisson Integral Formula for the Half-Plane

If is analytic in a domain containing the x-axis and the upper half-plane and in this domain, then the values of the harmonic function in the upper half-plane are given in terms of its values on the x-axis by (y>0). Below is an outline for the derivation, I just need to figure out how to justify the steps.

Quantitative Analysis in Business Decisions

Mr. Davison has been operating a small bicycle shop at the same location near Aspen, Colorado for 50 years. What type of decisions must he make in operating his business? On what basis would he likely be making these decisions? How do you think Mr. Davison would respond to a suggestion that he hire a quantitative analyst to assi

Complex Variables : Analytic Functions and Limits

Please help with the following mathematics-related problem. Let f(z) be analytic in a region G and setphi(z,w) = (f(w)-f(z))/(w-z) for w,z E G w does not equal z. Let z0 Ye G. Show that lim (z,w)-->(z0,z0) phi(z,w) =f'(z0). Complex Variables. See attached file for full problem description

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.

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

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:

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. &#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 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

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.

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

Open mapping theorem. Complex Analysis

Let P : C -> R be defined by P(z) = Re z; show that P is an open map but it is not a closed map. ( Hint: Consider the set F = { z : Imz = ( Re z)^-1 and Re z doesn't equal to 0}.) Please explain every step and justify.

Analytic functions complex

Let f = u + iv be an analytic function on an open connected set G in C ( C = complex plane) where u and v are its real and imaginary parts. assume u(z) >= u(a) for some a in G and all z in G. Prove that f is constant.

Power series representation of analytic functions (Complex integrals)

Evaluate the following integrals: a). integral over gamma of e^(iz) / z^2 dz, where gamma(t) = e^(it), 0=<t=<2 pi ( e here is exponential function). Please use basic definitions and power series representation of analytic functions to do so. b). integral over gamma of sin(z)/z^3 dz ( same gamma and values of t as abo