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Complex Analysis

Calculus

Question 1 Multiple Choice The two sides of a right triangle have lengths 2.92 and 3.98. Find the hypotenuse. □ 6.90 □ 3.34 □ 4.94 □ 3.20 Question 2 Multiple Choice An equation used in the study of protein molecule is In A+ In h - In(1 - h) Solve for

Complex Numbers : Polar Form and DeMoivre's Theorem

6. Please explain step by step Apply DeMoivre's Theorem to find (-1+i)^6 * change to polar form first You will recognize the angle, so put in the correct value of sine and cosine to reduce back to simple complex form 7. Find the fourth roots of 16(cos pi/4 + i sine pi/4) ; n=4 Please explain in detail 9 solve for

Complex Analysis and Function

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 for Holomorphic Functions

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 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