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Complex Analysis

Sum and difference identities

Find the coordinates of P(π/12) x = (1+√3)/(2√2) To find the y-coordinate you use the identity: sin(α - β) = (sinα)(cosβ) (cosα)(sinβ) Why do we use this identity? eg, why don't we use sin(α + β) = (sinα)(cosβ) + (cosα)(sinβ) ??

How do you solve for u(x,t) using separation of variables?

Consider a model of a damped, oscillating string of length L, u_u = -2(lambda)(u_e) + (c^2)(u)_zz over 0 <= x <= L, where u(x, t) is the displacement, lambda describes the damping and c is the natural (undamped) wave speed. Suppose that the ends of the string are fixed at u = 0, and that the string is initially at rest, b

Complex Variables

If a > e prove that the equation a*z^n=e^z has n solutions (counting multiplicities) inside of the circle |z|=1.

Prove that (a) &#9474; z1 &#9474;-&#9474; z2 &#9474; &#8804; &#9474;z1 - z2&#9474; &#8804; &#9474; z1 &#9474;+ &#9474; z2 &#9474; (b) &#9474; z1 &#9474;-&#9474; z2 &#9474; &#8804; &#9474;z1 + z2&#9474; &#8804; &#9474; z1 &#9474;+&#9474; z2 &#9474; where z1 , z2 are complex numbers.

Functions of a Complex Variables Prove that: (a) &#9474; z1 &#9474;-&#9474; z2 &#9474; &#8804; &#9474;z1 - z2&#9474; &#8804; &#9474; z1 &#9474;+ &#9474; z2 &#9474; (b) &#9474; z1 &#9474;-&#9474; z2 &#9474; &#8804; &#9474;z1 + z2&#9474; &#8804; &#9474; z1 &#9474;+&#9474; z2 &#9474;

Complex Analysis - Analytic Functions

Please answer the attached complex analysis questions. i.e. Prove the following generalization of proposition ... if g is analytic, if f is analytic

Complex variables 3.3.11

Please see the attached file for full problem description. --- Show all steps, even minor details Send response as attachment Provide common sense explanations Determine a branch of log that is analytic at z = -1, and find its derivative there.

Complex Variables 115: Bonus Question.

All steps must be shown, even small details. Workout the relevant calculations please complete the problem, not just set it up. Provide common sense explanations. Bonus: The function p(z)=[z(z+1)]^-1 can be written in two different ways: (see attached for full equation) These two expansions are contradictory. The first

Complex Variables Calculations.

All steps must be shown, even small details. Workout the relevant calculations please complete the problem, not just set it up. Provide common sense explanations. 6. For a>0, use residue calculus to evaluate (see the attachment for the full problem description and equation.)

Hint: First consider what curve z...

Compute: lim as r -> infinity |f(z)|, where z = re^[(i)("alpha")] Your answer will depend on "alpha". Hint: First consider what curve z = re^[(i)("alpha")] traces out in the complex plane as r -> infinity

Complex Variables. 1.2

What are all possible solutions of z^4 + 4 = 0? From this information, write out a complete factorization of z^4 + 4.

Solve a complex variable equation.

Z is a complex number, s and t are real numbers. Find z1 and z2 - the solution the solutions of the equation. (z^2)+(|z|^2)-(2Is)=(8t^2) in terms of s and t. If z1*z2=-8I find s,t

Calc II

Find the open interval of convergence and test the endpoints for absolute and conditional convergence.

Matlab help for solving non-linear equations

[note: suggested number of credits may be modified if necessary] Hi, Part (1/2) I need help/advice in solving non-linear equations using Matlab. My focus is on the - Bisection method - Regula falsi - Muller's method (if possible) I have seen programs for the above methods on the internet, but I could not see *e

Riemann Sum

Write out the Riemann Sum R(f,P, 1, 4), where f(x) = ln x, P = {1, 2, 2.4, 2.9, 3.4, 4} and ck is the midpoint of the interval [xk&#8722;1, xk] for each k. Get a decimal approximation for the Riemann Sum.

Help with an intro analysis proof

Due to the amount of symbols in the problem I attached a .pdf file. In words, I think the problem is asking - if the limit of f at c is positive, then on some deleted neighborhood of c, f is positive away from 0.

Poincare's Lemma and its Converse

Please see the attached file for the fully formatted problem. For  phi E C2[R3 ! R3], curl grad phi = 0. Prove this. The converse is "Poincare's Lemma": if f E C1[R3 --> R3] and if curl f = 0, then f is a gradient, i.e., f = grad  for some  2 C2. Try it this way: if f = grad phi, then phi (x1, x2, x3) = phi(0)+ ....

College Algebra

Can you please help me with these. I can not get these and I am trying to study for the clep exam. Do not do the circled problems only the ones listed below. Thank you for help. Please show step so that I can understand better. 129 1. a) 6, b) 12, c) 16, d) 20 2. a) 24, b) 26 3. a) 35, b) 36 4. a) 46, b) 48 P. 14

Pre calc

Need equation for: Ellipse Center at(0,4); focus at(8,4); vertex at (6,7)

Sequential Compactness

Please see the attached file for the fully formatted problems. Consider C[0, 1], the space of real valued continuous functions defined on the unit interval [0, 1]. Let K = C1[0, 1] {f : Z 1 0 f02  1, ||f||1  1} Note that C1[0, 1]  C[0, 1], and K  C[0, 1]. Show that K is compact in C. I am assuming compactness h