### Complex Variable Class - Undergraduate 500 Level

The beta function is this function of two real variables... Make the substitution t=1/(x+1) and use the result obtained in the example in Sec. 77 to show that... Please see attachment for equations.

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The beta function is this function of two real variables... Make the substitution t=1/(x+1) and use the result obtained in the example in Sec. 77 to show that... Please see attachment for equations.

Problem: Show that when ... Please see the attached file for the fully formatted problems.

Derive the Taylor Series representation ... (see attached)

This problems is from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution. Problem: Obtain the Taylor series ... (see attachment) for the function ... (see attachment)

Write z=re^i0, where 0<r<1, in the summation formula that was derived in the example in Sec. 52. Then, with the aid of the theorem in Sec. 52 show that... (See attachment for details)

In each case, determine the singular points of the function and state why the function is analytic everywhere except at those points: (a) f (z) = (2z + 1) / [z (z^2 + 1)] (b) f (z) = (z^3 + i) / (z^2 - 3^z + 2) (c) f (z) = (z^2 + 1) / [(z + 2)(z^2 + 2z + 2)]

9. Let f denote the function whose values are _ f (z) = z^2 / z when z ≠ 0, f (z) = 0 when z = 0. Show that if z = 0, then ∆w/∆z = 1 at each nonzero point on the real and imaginary axes in the ∆z, or ∆x∆y, plane. Then show that ∆w/ͧ

7. Prove that dzn-1/ dz = nzn-1, for the derivative of zn remains valid when n is a negative integer (n = -1, -2, ???), provided that z≠ 0. (Suggestion: Write m = -n and use the formula for the derivative of a quotient of two functions). See the attached file.

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

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.

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

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

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. 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.

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°)

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. 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)

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

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

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

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

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

Functions of a Complex Variables Analytic Functions If u = sin x . cosh y + 2cos x . sinh y + x2 - y2 + 4xy , then prove that u is a harmonic function and find the analytic funct

Functions of a Complex Variables ∞ ∞ Prove that │∑ zn │ ≤ ∑│zn │ where zn is a complex number. n =1 n =1

Please answer the attached complex variable problems. Thank you.

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

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

[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

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.

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