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Complex Analysis


Question 1 Multiple Choice The two sides of a right triangle have lengths 2.92 and 3.98. Find the hypotenuse. □ 6.90 □ 3.34 □ 4.94 □ 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

Quadratic equation

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). Questi

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.

Applying Complex Analysis and Argument Principle

Suppose f is analytic on B(bar) (0;1) and satisfies |f(z)| < 1 for |z| = 1. Find the number of solutions (counting multiplicities) of the equation f(z) = z^n, where n is an integer larger than or equal to 1. Please justify every step and claim and refer to any theorems you use.

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

Radius of Convergence and Abel's Theorem in Complex Analysis

A) I want to prove that if sum of a_n(z-a)^n have radius of convergence 1 and if the sum a_n converges to A then lim (r -> 1- ) of the sum (a_n r^n) = A. ( I believe z here is a complex number). B) Using Abel's theorem, prove that log2 = 1 - 1/2 + 1/3 - ...

I need help with these five algebra problems.

BACKGROUND INFORMATION: A simple pendulum, such as a rock hanging from a piece of string or the inside of a grandfather clock, consists of a mass (the rock) and a support (the piece of string). When the mass is moved a small distance away from its equilibrium point (the bottom of the arc), the mass will swing back and fort

Complex Analysis : Analytic Functions as Mappings

1). Let G be a region and suppose that f : G -> C ( C is complex plane) is analytic such that f(G) is a subset of a circle. Show that f is constant. 2). If Tz = (az + b)/(cz + d), find necessary and sufficient conditions that T(t) = t where t is the unit circle { z: |z| = 1}. My solution for number 2 is : T(t) = t , which

Complex Analysis : Mobius Transformation

1). Let D = {z: |z| < 1 } and find all Mobius transformations T such that T(D) = D. 2). Show that a Mobius transformation T satisfies T(0) = infinity and T ( infinity) = 0 if and only if Tz = az^-1 for some a in C ( C is complex plane).

Polynomial Equations : Complex Solutions, Conjugates and Shift Operator

Prove that if p is a polynomial with real coefficients, and if is a (complex) solution of P(E)z = 0, then the conjugate of z, the real part of z, and the imaginary part of z are also solutions. Note: This is from a numerical analysis course, and here P(E) refers to a polynomial in E, the "shift operator" for a sequence.

Analytic functions in complex plane

1). Determine the set A such that For r > 0 let A ={w, w = exp (1/z) where 0<|z|<r}. 2).Prove that there is no branch of the logarithm defined on G= C-{0}. ( C here is the complex plane). ( Hint: suppose such a branch exists and compare this with the principal branch). I want detailed proofs and please prove ever

Analytic Function Proofs and Real Functions

1) Show that the real part of the function z^(1/2) is always positive. 2) Suppose f: G --> C ( C complex plane) is analytic and that G is connected. Show that if f(z) is real for all z in G, then f is a constant.

Continuity complex plane

Let G be an open subset of C ( complex plane) and let P be a polygon in G from a to b. Use the following 2 theorems to show that there is a polygon Q in G from a to b which is composed of line segments which are parallel to either the real or imaginary axes. The 2 theorems are: 1). Theorem: Suppose f: X --> omega is continuou

Metric spaces and the topology of complex plane

Show that { cis k : k is a non-negative ineger} is dense in T = { z in C ( C here is complex plane) : |z| = 1 }. For which values of theta is { cis ( k*theta) : K is a non-negative integer} dense in T ? P. S. cis k = cos k + i sin k, i here is square root of -1. I want a full justification for each step or claim.

Stereographic projection on complex plane

Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p / S = V ( / = intersection). Recall from analytuc geomerty that P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}. Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_3)

DeMoivre's Theorem and Power of Complex Numbers

See the attached file for complete equations --- Use DeMoivre's Theorem to find the indicated power of the following complex numbers: 1. Find the fourth roots of 256(1+&#61654;3i ) 2. Find all solutions of the equation x3 &#61485; 27i = 0 [please show all the steps, including the algebraic ones] ---

Harmonic Functions and Cauchy-Riemann Equations

15. Determine whether or not the given function u is harmonic and, if So, in what region. If it is, find the most general conjugate function v and corresponding analytic function f (z). Express f in terms of z. Please see the attached file for the fully formatted problems.