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# Complex Analysis

### 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

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 &#960;/6 &#8804; x &#8804; &#960;/4 I think I have to use substitution for the integration

### Imaginary Powers / Residues : Allowing Functions

1. Allowing z = x + iy, find all of the roots for z^i=-2i. 2. By evaluating residues only, solve integral (-infinity --> infinity) xsinx/(x2 -2x =2)^2 dx

### 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.

### Complex Analysis and Singularities : If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

### Complex Cross Ratio : Evaluate cross ratio

Evaluate cross-ratio (infinity,0,i,1) give answer in the form a + ib where a,b in R.

### Complex / Entire Function : Let f be an entire function such that |f(z)| =<10|z+1| for all |z|>100. Show that f is a linear function, f(z)= pz + q.

Let f be an entire function such that |f(z)| =<10|z+1| for all |z|>100 Show that f is a linear function, f(z)= pz + q.

### 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.

### Concept of squared variables in a system of equations

Kindly elaborate on the concept of squared variables in a system of equations by the sample that was given to me. See attached file for full problem description.

### 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

### Let G be a region and suppose that f:G->C (C here is complex plane)is analytic and a in G such that |f(a)|=<|f(z)| for all z in G. Show that either f(a) = 0 or f is constant.

Let G be a region and suppose that f:G->C (C here is complex plane)is analytic and a in G such that |f(a)|=<|f(z)| for all z in G. Show that either f(a) = 0 or f is constant.

### Potential Flow Theory : Conformal Transformation and Velocity on the Surface of an Ellipse

A) Show that if b<a the conformal transformation....maps a circle of radius 'a' in the plane into an ellipse in the ...plane. b) Show that the velocity on the surface of an ellipse in a uniform horizontal flow of velocity U reaches a maximum when theta = 90 degrees and has a magnitude of... c) Determine the velocity on the sur

### 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... ---

### Show that SO(4) is isomorphic to the quotient

Show that SO(4) is isomorphic to the quotient of SU(2) X SU(2) by the subgroup generated by (-1,1).

### 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

### Finding the side of base of a pyramid from surface area.

The surface area of the right squre pyramid is given by S=b square rt. b^2 + 4h^2. If the pyramid has height of 10 feet and surface area of 100 square feet, what is the length of a side b of its base?

### 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.

### Forming Relationships between Variables

Please help! I cannot seem to get this problem solved! (See attached file for full problem description)

### 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