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Complex Analysis

Complex Expressions: Cartesian and Polar Form

For questions 1 - 3 give answer in Cartesian form and question 4 in polar form. Q1 2z1 + z2 - 4z3 where z1 = 5 - j7, z2 = 4 + j, z3 = 8 - j5 Q2 z1z2 where z1 = - 3 - j5, z2 = 4 + j5 Q3 where z1 = 1 + j6, z2 = 3 + j7 Q4 Please see the attached file for the fully formatted problems.

Calculus Critical Point Derivatives

1. Given f(x,y) = x^2-4xy+y^3+4y Find the critical points and then use the Saddle Point Derivative Test to determine if they are max, min, or saddle points. 2. Given f(x,y)=4xy-x^4-y^4 Find the critical points and then use the Saddle Point Derivative Test to determine if they are max, min or saddle points. 3. Find the

Algebra & Complex Numbers : Amplitude Ratio

Make y the subject of the formula E = p(1-e^(y-1)) If the amplitude ratio, N in decibels is given by n = 10log(P0/P1) and the power is given by P=(V^2/R), show that for matched input and output resistances the output Vo is related to input voltage Vi by Vo = Vi 10^(N/20) If N is increase by 6 dB, show that output volta

Quadratic equation

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Questi

Open mapping theorem. Complex Analysis

Let P : C -> R be defined by P(z) = Re z; show that P is an open map but it is not a closed map. ( Hint: Consider the set F = { z : Imz = ( Re z)^-1 and Re z doesn't equal to 0}.) Please explain every step and justify.

Proof of absolute maximum and minimum

I would like help with the following problem: Find with proof the absolute maximum and minimum values of f(x) = x^4 + 2x^2 - 4 on the interval [0,3]. There's a hint saying that you can prove this using the mean value theorem. Thanks for all of your help.

Average value on an interval

Find the average value of y=csc2xcot2x on the interval π/6 ≤ x ≤ π/4 I think I have to use substitution for the integration

Applying Complex Analysis and Argument Principle

Suppose f is analytic on B(bar) (0;1) and satisfies |f(z)| < 1 for |z| = 1. Find the number of solutions (counting multiplicities) of the equation f(z) = z^n, where n is an integer larger than or equal to 1. Please justify every step and claim and refer to any theorems you use.

Complex Analysis / Singularities / Argument Principle

Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G. In your solution, please refer to theorems or certain lemmas. Justify your claims and steps. I want to learn not just have the right answer. Thanks.

Complex Analysis / Singularities

One can classify isolated singularities by examining the equations: lim (z -> a) |z - a|^s |f(z)| = 0 lim(z -> a) |z - a|^s |f(z)| = infinity Now, prove that a function f has an essential singularity at z = a iff neither of the above holds for any real number s.

Analytic functions complex

Let f = u + iv be an analytic function on an open connected set G in C ( C = complex plane) where u and v are its real and imaginary parts. assume u(z) >= u(a) for some a in G and all z in G. Prove that f is constant.

Power series representation of analytic functions (Complex integrals)

Evaluate the following integrals: a). integral over gamma of e^(iz) / z^2 dz, where gamma(t) = e^(it), 0=<t=<2 pi ( e here is exponential function). Please use basic definitions and power series representation of analytic functions to do so. b). integral over gamma of sin(z)/z^3 dz ( same gamma and values of t as abo

Root of the Problem Descriptions

(See attached file for full problem description with proper symbols and equations) --- First: solve this problem. Second: check my answer. Third: if my answer is wrong or incomplete explain why. Explain why cannot have more than one root. This is how I tried to solve it: I used the interval [-1,1] By the

Potential flow theory

(See attached file for full problem description) --- The complex potential of a two-dimensional motion is... ---

Potential flow theory

(See attached file for full problem description) --- The complex potential for a flow over a body is given by... ---

Radius of Convergence and Abel's Theorem in Complex Analysis

A) I want to prove that if sum of a_n(z-a)^n have radius of convergence 1 and if the sum a_n converges to A then lim (r -> 1- ) of the sum (a_n r^n) = A. ( I believe z here is a complex number). B) Using Abel's theorem, prove that log2 = 1 - 1/2 + 1/3 - ...

I need help with these five algebra problems.

BACKGROUND INFORMATION: A simple pendulum, such as a rock hanging from a piece of string or the inside of a grandfather clock, consists of a mass (the rock) and a support (the piece of string). When the mass is moved a small distance away from its equilibrium point (the bottom of the arc), the mass will swing back and fort

Complex Analysis : Analytic Functions as Mappings

1). Let G be a region and suppose that f : G -> C ( C is complex plane) is analytic such that f(G) is a subset of a circle. Show that f is constant. 2). If Tz = (az + b)/(cz + d), find necessary and sufficient conditions that T(t) = t where t is the unit circle { z: |z| = 1}. My solution for number 2 is : T(t) = t , which

Complex Analysis : Mobius Transformation

1). Let D = {z: |z| < 1 } and find all Mobius transformations T such that T(D) = D. 2). Show that a Mobius transformation T satisfies T(0) = infinity and T ( infinity) = 0 if and only if Tz = az^-1 for some a in C ( C is complex plane).

Complex Numbers : RSA Cipher and Summation

A=1,B=2,C=3,D=4,E=5,........X=24,Y=25,Z=26 It is enciphered using the rule 35 R (m)=m (mod 91) 35 The resulting ciphertext is ( 73,14,23,73,23) Verify the rule given satisfy the condition for an RSA cipher. Using the repeated squaring technique decipher the ciphertext and find the message