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Complex Analysis

Complex Variables : Limits

11. T (z) = (az + b) / (cz + d) (ad - bc ≠ 0) Show the following. (Please explain by using theorem.) (a) lim T (z) = ∞ if c = 0 z→∞ (b) lim T (z) = a / c and lim T (z) = ∞ if c ≠ 0. z→∞ z→-d/c

Complex Variables : Limits

10. Show the following limits. (Please explain by using theorems.) (a) lim 4z2 / (z - 1)2 = 4 z→∞ (b) lim 1 / (z - 1)3 = ∞ z→1 (c) lim (z2 + 1) / (z - 1) = ∞ z→∞

Comples Variables : Limits

5. Show that the limit of the function _ f (z) = ( z / z )2 as z tends to 0 does not exist. Do this by letting nonzero points z = (x, 0) and z = (x, x) approach the origin. (Note that it is not sufficient to simply consider points z = (x, 0) and

Simplify a Complex Expression

Suppose that f (z) = x2 - y2 - 2y + iּ(2x - 2xy), where z = x + iy. Use the expressions _ _ x = (z + z) /2 and y = (z - z)/2i to write f (z) in terms of z, and simplify the result.

Complex Variables: Verify Inequality

3. Verify that (sqrt(2))ּ|z| ≥ |Re z| + |Im z|. Suggestion: Reduce this inequality to (|x| - |y|)2 ≥ 0. Sqrt(2) means square root of 2.

Kirchoff's Laws : Mass-Spring Equation

Consider a basic electric circuit with a resistor, capacitor, and inductor and input voltage V(t). It follows Kirchoff's Laws that the charge on the capacitor Q = Q(t) solves the differential equation: {see attachment}, where L (inductance), R (resistance), and C (capacitance) are positive constants (depending on material). The

Convolution Applied to Inverse Transform Problem

Please see the attached file for the fully formatted problem. My problem lies in manipulating the equation to one which the inverse transform can be taken, but would appreciate this example of convolution worked with some more of the blanks filled. I cannot figure out how the equation gets manipulated into what appears

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