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    Complex Analysis

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    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→∞

    Complex 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

    Complex Variables : Limits

    3. Let n be a positive integer and let P(z) and Q(z) be polynomials, where Q(z0) ≠ 0. Find the following limits. (Please explain by using relevant theorem.) (a) lim 1/ zn (z0 ≠ 0) z→z0 (b) lim (iz3 -1) / (z + i) z→i (c) lim P(z) / Q(z) z→z0

    Complex variable

    Please specify your notation(if necessary) and explain clearly each step of your solution. Thank you very much. 7. Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation _ = ez , and label corresponding portions of the boundaries.

    Complex variables example problems

    Please specify your notation(if necessary) and explain clearly each step of your solution. Thank you very much. Sketch the region onto which the sector r ≤ 1, 0 ≤ θ ≤ π/4 is mapped by the transformation (a) ω = z2 (b) ω = z3 (c) ω = z4

    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.

    De Moivre's Theorem and Rectangular Coordinates

    5. Use de Moivre's formula to derive the following trigonometric identities. (a) cos 3θ = cos3 θ - 3cos θּsin2 θ (b) sin 3θ = 3cos2 θּsin θ - sin3 θ 6. By writing the individual factors on the left in exponential form, performing the needed operations, and finally changing back to rectangular coordinates, show tha

    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.

    Associative and Commutative Laws for Multiplication

    1. Use the associative and commutative laws for multiplication to show that: (z1z2)(z3z4) = (z1z3)(z2z4) 2. Prove that if z1z2z3 = 0, then at least one of the three factors is zero. Please see attachment for proper citation of equations.

    The problems are from Boundary Value Problems

    The problems are from Boundary Value Problems. Undergrad 400 level course. Mainly uses partial differential skills. Some problems might require using MATLAB. Please explain each step of your solutions. Thank you very much.

    Complex Number in Trigonometric Form

    Write the complex number in trigonometric form r(cos θ + i sin θ ), with θ in the interval [0°, 360°). 2√3 - 2i Which is the correct answer? 4( cos 30° + i sin 30°) 4( cos 330° + i sin 330°) 4( cos 60° + i sin 60°) 4( cos 300° + i sin 300°)

    Multiplication of Complex Numbers

    Find the product. Write the product in rectangular form, using exact values. [8 cis 210°] [6 cis 330°1] Which is the correct answer? a) 24i b) -12 + 12 x square root of 3i c) -48 d) 12 x square root of (3 + 12i)

    Write the complex number in rectangular form.

    Write the complex number in rectangular form. cis 210° Which is the correct answer? a) -square root of 3/2 + i1/2 b) -1/2 + i(-1/2) c) square root of 3/2 + i(-1/2) d) -square root of 3/2 + i(-1/2)

    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 t

    Fields : Complex Numbers

    LET F = a b -b a : a,b is an element of C (complex numbers), a² + b² ≠ 0 Is (F, +, .) a field?

    Quadratic Equations and Applications (Four Problems)

    Assignment 5: Solutions of Quadratic Equations and their Applications 1. a) x2 + 6x + 7 = 0 b) z2 + z + 1=0 c) (3) ½ Y2 4y-7 (3) ½ = 0 d) 2x2 - 10x + 25 = 0 e) 2x2 +6x + 5 = 0 f) s2 - 4s + 4 = 0 g) 5/6x2 - 7x - 6/5 = 0 h) 7a2 +8a + 2 = 0 2. Given x = 1 and x = -8, form a quadratic equation. 3. What ty

    Real, Rational and Complex Numbers

    Please see the attached file for the fully formatted problems. 1. Classify the given numbers as real and rational, real and irrational, or complex. 2. a. Select any irrational number, and turn it into a rational number by using addition, subtraction, multiplication, division, or exponentiation. b. Select any imagina

    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.

    Inequalities of Complex Variables.

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

    Branch of logarithmic equation and derivative question

    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.