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Complex Analysis

Complex analysis

Differentiability (See attached file for full problem description) --- Find all functions f=u+iv which are differentiable everywhere and which have: ? u(x,y)=y ? u(x,y)= ---

Complex analysis

Holomorphic. (See attached file for full problem description) --- 1) Let . ? Show that f is holomorphic in . ? Find its derivative f' and show that f' is holomorphic in . 2) Define a function f by where . Determine all points z where f is differentiable. ---

Complex and Real Solutions

Let A be a complex number and B a real number. Show that the equation |z|^2+Re(Az)+B=0 has a solution if and only if |A|^2 >= 4B. If this is so, show that the solutions set is a circle or a single point.

Conversion of Complex Numbers to Cartesian and Polar Form

Convert each of the following to polar form. 1. 9 - j5 giving the argument in Radians 2. 9 + j16 giving the argument in Degrees Convert each of the following to polar form 2. pi / 7 Please see the attached file for the fully formatted problems.

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 3 Review

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

Alegebra & Complex Numbers : Amplitude Ratio and Input and Output Voltage

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

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.

Real Analysis / Absolute Max and Min

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.

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

(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

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

Polynomial Equations : Complex Solutions, Conjugates and Shift Operator

Prove that if p is a polynomial with real coefficients, and if is a (complex) solution of P(E)z = 0, then the conjugate of z, the real part of z, and the imaginary part of z are also solutions. Note: This is from a numerical analysis course, and here P(E) refers to a polynomial in E, the "shift operator" for a sequence.

Analytic functions in complex plane

1). Determine the set A such that For r > 0 let A ={w, w = exp (1/z) where 0<|z|<r}. 2).Prove that there is no branch of the logarithm defined on G= C-{0}. ( C here is the complex plane). ( Hint: suppose such a branch exists and compare this with the principal branch). I want detailed proofs and please prove ever