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Complex Analysis

Complex Variables : Region of Flow, Fluid Presure and Speed of Fluids

4. Show that the speed of the fluid at points on the cylindrical surface in Example 2, Sec. 108. (below) is 2A| sin θ| and also that the fluid pressure on the cylinder is greatest at the points z = ±1 and least at the points z = ± i --- - Show the speed of the fluids... - At an interior point of a region of flow...

Four complex variable problems (87-4,5,6,10)

See the attachment for the problems --- - Find the bilinear transformation that maps... - Show the disposition of two linear fractional transformations... - A fixed point of transformation w = f(z)... - Show that there is only linear fractional transformation that maps... --- (See attached file for full problem descri

Four complex variable questions (83-5, 85-5, 7, 13)

See the attachment for the problems. --- - Find the region onto which the half plane y>0... - Find the image of the quantrant x>1, y>0... - Describe geometrically the transformation w=1/(z-1). - Using the exponential formz = re^(i x theta) of z... ---- (See attached file for full problem description)

Decision Analysis with Payoff Tables and States of Nature

A payoff table is given as States of Nature .... a. What decision alternative would be made using a minimax regret? b. What decision alternative would be made by using a conservative approach c. What decision alternative would be made by using a maximax approach?

Decision Analysis with Payoff Tables and States of Nature

A payoff table is given as ..... a. What choice should be made by the optimistic decision maker? b. What choice should be made by the conservative decision maker? c. What decision should be made under minimax regret? d. If the probabilities of E, F, and G are .2, .5, and .3, respectively, then e. What choice should be ma

Diffusion Equations : Separation of Variables

4. (Separation) Seeking a solution u(x, t) = X(x)T(t) for the given PDE, carry out steps analogous to equations (3)?(6), and derive ODE's analogous to (7a,b). Take the separation constant to be ?K2, as we do in (6). Obtain general solutions of those ODE's (distinguishing any special ic values, as necessary) and use superposition

Complex Conjugates

What is the complex conjugate of 78.93iw^3+30.48iw^2+iw? How did you find it?

Weighted Voting System

1) For each weighted voting system , find all dictators (d), veto power players( vp), and dummies (d) ( 7: 7,3,2,1) 2) For the weighted voting system ( 12: 6,4,3,1,1,) a) Find what percent of the total vote is the quota. b) In a Shapley-Shubik distribution system , how many sequential coalitions would be formed from thi

Conformal Mapping

Find a conformal mapping from the unit disc &#916;(0,1) = {z|z|<1} to D={z:|z|<1}[0,1]. Please see attached for diagram.

Cauchy's Formula

Use Cauchy's formula for the derivative to prove that if f is entire and |f(z)|&#8804; A|z|² + B|z| + C for all z&#949;C, then f(z) = az² +bz + c Please see attached for full question.

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.

Complex Variables : Taylor Series

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)

Determine the singular points of the function

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)]

Complex Variables : Differentiability

9. Let f denote the function whose values are _ f (z) = z^2 / z when z &#8800; 0, f (z) = 0 when z = 0. Show that if z = 0, then &#8710;w/&#8710;z = 1 at each nonzero point on the real and imaginary axes in the &#8710;z, or &#8710;x&#8710;y, plane. Then show that &#8710;w/&#871

Complex Variables: Differentiation - Quotient Formula

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.

Complex Variables : Limits

3. Let n be a positive integer and let P(z) and Q(z) be polynomials, where Q(z0) &#8800; 0. Find the following limits. (Please explain by using relevant theorem.) (a) lim 1/ zn (z0 &#8800; 0) z&#8594;z0 (b) lim (iz3 -1) / (z + i) z&#8594;i (c) lim P(z) / Q(z) z&#8594;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 &#8805; 0, 0 &#8804; y &#8804; &#960; 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

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

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 &#952; + i sin &#952; ), with &#952; in the interval [0°, 360°). 2&#8730;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°)