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# Complex Analysis

### 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

### Standard Complex Form

Perform the indicated operations. Write the answer in the form a + bi. 2+4i (9+4i)

### 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 origin 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

### Roots Real, Repeated Real, or Complex?

Determine if the following equations have a solution or not? Justify your answer. Are the roots real, repeated real or complex? Q:) if x=3 and x= -5, then form a quadratic equation.

### 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

### Quantitative Methods-Integer Variables

Briefly explain why the use of integer variables creates additional restrictions but provides additional flexibility.

### 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

### Rearranging a Complex Expression

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. R1 - j &#969;C = R2 + R3 R4 + R5 R6 + j &#969;L

### 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

### Functions: Inverses and Composite Functions

The problems are all in multiple choice format so pick the most correct answer in each case and show your work. The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and + 4i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients. a. f(x) =

### Complex Metric Spaces

Let (S,d) be a metric space and define the function u(x,y) = d(x,y)/(1+d(x,y)) for all x,y in S. (a) Prove that u is a metric on S with sup u(x,y) <= 1. (b) If S = C (complex) and d is the usual Euclidean metric d(z,w) = abs(z-w), then prove that sup u(z,w) = 1. (c) For 0 < r < 1, show that u(x,y) < r if and only

### 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.

### Complex Proof : Analytic Functions

Let f be an entire function such that |f(z)| &#8804; A|z| for all z in C for some fixed positive real number A. Use the attached theorem to show that f(z) = mz for some complex number m.

### 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=?

### Principal Branch and Complex Powers

How do you find (and what is it) the principal branch of the following complex powers: (e/2)(1-i(sqrt(3)))^(3pi*i) and (1-i)^(4i).

### Sample 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 Variables : Taylor Polynomial Proof

Please see the attached file for the fully formatted problems.

### 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

### Complex Numbers and the Roots of an Equation

If the solution to an equation is imaginary or irrational, it takes a bit more effort to check. Replace x by each given number to verify the following statement. Both 2+3i and 2-3i satisfy x^2 -4x+13=0

### Solve y < -2 + 7 Functions

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

### Greatest Common Divisor ( GCD ) using the Euclidean Algorithm

Using the gcd for 85 and 1+13i, by the Euclidean Algorithm.

### Write a cis expression in standard complex form (a+bi)

Use Demoivre's theorem to write [3 cis(pi/2)]squared in the form a+bi without trigonometric functions. Find the real and complex solution of the equation Xsquared - 8 = 0 Find the angle between the two vectors (5,2) and (-2,5)

### Binomial Expansion

What is the coefficient of x5 in exp anion of (x+3)8

### 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 : Analytic Functions and Laurent Series

Any insight on where to go with these problems would be helpful and appreciative. This is my first time in complex analysis and I am having problems understanding the concepts. Seeing solutions to problems is helping me to understand how to approach other problems. Thank you very much! I need help specifically on problems 2