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Complex Analysis

Complex Analysis refers to the study of complex numbers. In dealing with Complex Analysis, it is important to understand the different terms. A complex number, in Mathematics, is a number that can be expressed in the form a+bi, where ‘a’ and ‘b’ are real numbers, and ‘i’ is the imaginary unit. It is also important to note that i^2=-1. Thus, a+bi is a complex number because it is a mix of real and imaginary, with the ‘a’ being real, and the ‘b’ being imaginary. Complex numbers are often depicted on a plot, where the x-axis represents the real value of the complex number (a) and the y-axis represents the imaginary value of the complex number (b). The point, which is plotted, is thus known as point (a, b). One thing to consider is that numbers written in the a+bi form are not necessarily complex numbers. For example, if a = 0 then the number is purely imaginary. If b = 0, then the number is completely real.

The creation of these complex numbers gives an additional tool to solve problems that cannot be solved with just real numbers alone. However, one important realization to note is that these imaginary numbers are no more or less fictitious than any other kind of number. With this in mind, these complex numbers have practical applications in many fields other than Mathematics – most prominently in electrical engineering. For example the imaginary ‘I’ is sometimes used to designate current in power systems. AC circuits in particular have their resistance and reactance denoted by two complex numbers. The sum of these two complex numbers is then known as the impedance, denoted by the symbol Z.

Thus, understanding complex numbers, and by extension Complex Analysis is a crucial skill to possess when dealing with complex problems.

Financial statement analysis...

Financial statement analysis Below are links to the annual reports for Microsoft, Adobe and Oracle: Microsoft Corporation is a well-known company engaged in developing, licensing and supporting a range

Complex Numbers and Analytic Functions

Prove algebraically that for complex numbers, |z1| - |z2| (is less than or equal to) |z1 + z2| (is less than or equal to) |z1| + |z2| Interpret this result in terms of two-dimensional vectors. Prove that |z-1| < |sqrt(z^2 - 1) < |z + 1|, for H(z) > 0. Show that complex numbers have square roots and that the square roo

Computing integrals via analytic continuation

Using the identity: Integral from 0 to infinity of x^(-q)/(1+x) dx = pi/[sin(pi q)] valid for 0 < q < 1 Compute the integral: Integral from 0 to infinity of x^p/(x^2+4 x + 3) dx for -1 < p < 1

Solving Difference Equation and Differential Equation

1. Complex Exponentials: Simply the following expression and give your answer both in polar and rectangular form. o c=3ejπ/4+4e−jπ/2 2. Difference Equations: Solve the following difference equation using recursion by hand (for n=0 to n=4) o y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0 3. Differential Equations

Contour Integration & Cauchy Theorem

Please show all your work. Find the possible values of: (Please see the attached file for the complete problem description). Where (please see the attached file) is a closed contour not passing through (please see the attached file) Thanks

Complex variable ( Rouche's Theorem)

I need some help showing a step-by-step calculation to this question: (better formula representation in attachment): Astronomer Cal Sagan Discovered from smoke filled telescopic lenses, a planet with billions upon billions of moons call that number N moons, including multiplicity, which he termed "fuzzy" moons. He came up with

complex variable res/sin.

1. Calculate residue for a complex function 2. Prove a meromorphic function on an extended complex plane is a rational function If f has a pole of order M at p then: (see attached) using the above: compute the residues at each singularity in the complex plane C for the following function see attached

Singularities of Analytic Functions

Please see the attached file for the functions. Question: For the following function, determine the nature of the singularities (removable, pole, or essential) and calculate the residues. For pole singularities, calculate the multiplicity as well.

Complex Integrals - Cauchy's Integration Formula

Three questions to do with Cauchy's integral formula are evaluated in clear, easy to understand steps. 1. Show that see attached where C is the semicircule |x|=R>1, lm(x)>=0 from x=-R to z=R. 2. Use Cauchy's integral formula for derivatives to evaluate see attached where |z-2|=2 is oriented clockwise. 3. Evaluate

Questions on Linear Fractional Transformation (LFT)

** Please see the attached file for the full problem description ** I want someone to show me the calculations and explain where necessary. Thanks in advance. 1) (i) Write down the equation of the line in xy-coordinates defined by | z - (i + 1) | = | z + 2i |. (ii) The LFT w = 1/z takes the line in (i) to a

Calculus and Volume question

From a rectangular sheet of metal measuring 120mm by 75mm, equal squares of side x are cut from each of the four corners. The remaining flaps are then folded upwards to form an open box. a) Draw a neat and simple diagram of the rectangular sheet of metal and show the dimensions (not drawn to scale) given including the squares

Complex Function Calculus

1. Let C denote the positively oriented circle |z|=2 and evaluate the integral .. see attachment 2. a) Fine the bi-linear transformation w=S(z) that maps the crescent-shaped region that lies inside the disk D:|z-2|<2 and outside the circle |z-1|=1 onto a horizontal strip b) Graph 3... see attachment

Laurent series expansion of complex functions

For each of the functions f(z) find the Laurent Series expansion on for the given isolated singularity (specify R). Then classify as an essential singularity, a pole (specify the order), or a removable singularity. Then find See Attached

The solution gives detailed steps on determining the maximum area for a specific question. All formula and calculations are shown and explained. Last, a graph using winplot software is given.

Include a copy of the description of the problem (as seen below) with your submission. Numerical, Graphical, and Analytic Analysis. The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation xof the sides so that the area of the cros

Series and singularities of complex functions

1. Find the series representations for the function f(z) = z/[(z-2)(z^2-1)] in powers of z that is valid when 1) |z| < 1 2) 1 < |z| < 2 3) |z| > 2 2. Fine all singularities of the function f in C, and determine whether it is a pole (find its order), a removable singularity, or an essential singul

Complex variables: Contour interval, Taylor series

** Please see the attachment for the full equation ** Find the contour integral (please see the attached file) where [gamma] is the circle |z| = 4 oriented counterclockwise. Find the Taylor series of the function f at z_0 and the derivative f^(50) (z_0) when: 1) f(z) = (2z+1)/(z(z-2) at z_0 = 1 2) f(z) = sin z at z_0 =

Calculations with Complex Variables

The questions in the attachments are regarding calculations with complex variables and cover concepts including Cauchy-Reimann and constant functions. Please provide detailed step-by-step calculations and explanations.

Topology of Complex Numbers

Assume S is a subset of the complex number set 1. an interior point of S 2. an exterior point of S 3. a boundary point of S 4. an accumulation point of S Please make the explanations clear and give examples.

Properties of Complex Variables

Detailed step by step calculations of the attached questions regarding complex variables including the domain, limits and continuity of complex functions. 1. For each of the functions below, describe the domain of definition, and write each function in the form f(z) = u(x,y) +iv(x,y) 1) f(z) = z^2 / (z+z) 2) f(z) = z^3 3

Dimensional Analysis Technique

Please use Dimensional analysis technique and show your work. 1-What is the length in meters of a 300. ft football field? 2-If a professional basketball player was 6.75 feet tall, what would be his equivalent height in centimeters? 3-A sheet of standard U.S typing paper measures 8.50 inches x 11.0 inches. What are these dim

Proving Complex Variables

** Please see the attached file for the full problem description ** Please show all work in detail and how all steps. 1. Prove that : ((2)^.5)*|z| ≥ |Re(z)| + Im(z) 2. Suppose that either |z| =1 or |w|=1. Prove that |z-w| = |!-z(bar)*w| 3. Prove that |z|=0 iff z=0

Incremental Analysis and Project Addition

Sales................................................................$950,000 Variable costs.................................................. 450,000 Fixed costs....................................................... 310,000 A proposed addition to Farrell's factory is estimated by the sales manager to increase sales by a

Ideal of continuous functions vanishing at a point

Let C[a,b] be the ring of complex valued continuous functions on closed and bounded interval [a,b]. You may assume that it is a ring under the operation of addition and multiplication of functions. Show that I_c = {f|f(c)= 0} is an ideal in this ring.

Data Analysis: Standard Deviation

I think that I did the first part right I just need reassurance, but I am really having difficulty with finding the percentage. Do you think that you could help? Using the sample data here 55, 52, 76, 40, 50, 65, 40. State the mode, Find the median, and state the mean. The sample standard deviation is 13.0. What percent

Statistical Analysis in a Random Sample

You select a random sample of n=14 families in your neighborhood and find the following family size (number of people in the family). 6 7 10 9 8 6 7 7 6 7 6 7 8 9 a) What is the mean family size for the sample? I think the answer here is all the above #'s * 14=7.37143 The mean family size for the sample is _7.36

Minimum Sample Size Analysis

One researcher wishes to estimate the mean number of hours that high school students spend watching TV on a weekday. A margin of error of 0.25 hour is desired. Past studies suggest that a population standard deviation of 1.7 hours is reasonable. Estimate the minimum sample size required to estimate the population mean with the s

Complex Variables and Applications

f(z) is defined by the equations: f(z)=1 when y<0 and f(z)=4y when y>0, and C is the arc from z=-1-i to z=1+i along the curve y = x^3. The answer is given as 2+3i. For C1 I get z=-x-ix^3, but the book says it is x+ix^3 (-1<x<0).