### Complex Variables : Limits and Differentiability

3. Give a direct proof that f ΄(z) = -1 / z2 when f (z) = 1 / z (z ≠ 0). Use this definition to prove the problem. dw / dz = lim ∆w / ∆z ∆z→0

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3. Give a direct proof that f ΄(z) = -1 / z2 when f (z) = 1 / z (z ≠ 0). Use this definition to prove the problem. dw / dz = lim ∆w / ∆z ∆z→0

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

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→∞

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

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.

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.

Calculate the square root of i using a direct method and show that using Euler's formula yield the same result.

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

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

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) ??

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: (a) │ z1 │-│ z2 │ ≤ │z1 - z2│ ≤ │ z1 │+ │ z2 │ (b) │ z1 │-│ z2 │ ≤ │z1 + z2│ ≤ │ z1 │+│ z2 │

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

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

Please answer the attached complex variable problems. Thank you.

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.

Find the limit of f(z) = x^2/(x^2+y^2) +2i Where z=x+iy and |z| --> 0

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

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.)

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

Please see the attached file for full problem description.

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

Please see attachments

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.

Find (1+i)^100

[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

Write out the sum ∑7,k=3 (k2 − 3k + 1) and calculate it.