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Complex Analysis

Complex numbers

1. Find all the values of z in the form a+bi such that (a),(b),(c) (please see the attachment) 2. Find the real part u(x,y) and determine if it is harmonic. (please see the attachment)

Complex analysis

Let f be analytic inside and on the unit circle. Suppose that 0<|f(z)|<1 if |z| = 1. Show that f has exactly one fixed point inside the unit circle. ( note : a fixed point is a point Zo such that f(Zo) = Zo).

Combinatorial Analysis

I have attached chapter 1 description pages. I need following problems. Page 16 : - Problem Number: - 10 Page 17: - Problem Number: - 15 Page 19: - Problem Numbers :- 5 and 8 Please mention each and every step. Thanking you

Mathematicians say that division by zero is forbidden.

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

Complex Inner-Product Space : Complex Spectral Theorem

Suppose that V is a complex (i.e. F = C) inner-product space. Prove that if N, an element of L(V), is normal and nilpotent, then N = 0. Use Complex Spectral Theorem: Suppose that V is a complex inner-product space and T is an element of L(v). Then V has an orthonormal basis consisting of eigenvectors of T if and only if T

Parabolas and Complex Roots

How many times does the graph of a quadratic equation (a parabola) cross the x-axis? I know that the graph of a quadratic equation should cross the x-axis 2 times, because the polynomial equation of second degree should have two roots, however if the vertex of a parabola is the orgin itself, the two points are coincident,

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.

Equation

I need to find all real or imaginary solutions to this equation. Please see attached

Evaluating complex numbers

Given the complex numbers z1 = 5 - j4 z2 = 4 + j z3 = -6 - j7 z4 = j2 Calculate, giving your answers in the form a + jb, the following:- (i) z4 - z1 + z2 (ii) 3z1 - 2z3 + z4 (iii) z1z2 (iv) z3/z2

Solve complex equation

In the following equation by equating real and imaginary parts, find expressions for R6 and L in terms of R1, R2, R3, R4, R5 and C, given that the frequency is one radian per second. See attached file for full problem description.

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

Systems of Complex Equations

x ------------------ = 0.007 754 1 + i --------- y 42425 x -------------------------------------- = 1 754 (1+ i -------) *500* (1 + i 41.47) y USING THESE TWO EQUATIONS SOLVE FOR X=? AND Y=?

Mobius Transformations

Prove that if T is a Mobius transformation such that T(0) = 0, then T may be written as T(z) = z/(cz+d) for some choices of c and d.

Complex Number Form : a+bi

Write each expression in the form of a+bi, where a and b are are real numbers 1. -3i/3-6i 2. -2-sqrt-27/-6

Solve y < -2 + 7

Y < -2 + 7 i need help to solve this equation and the steps on how to solve it.

Complex Variables : Analytic Functions and Limits

Let f(z) be analytic in a region G and set &#966;(z,w) = (f(w)-f(z))/(w-z) for w,z &#1028; G w &#8800; z. Let z0 &#1028; G. Show that lim (z,w)-->(z0,z0) &#966;(z,w) =f'(z0). Complex Variables. See attached file for full problem description