### Complex Cross Ratio : Evaluate cross ratio (infinity, 0, i, 1) give answer in the form a + ib where a,b in R.

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

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

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 = 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.

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.

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

(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

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

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

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).

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

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.

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

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

Find the solution to the given system for the given initial condition x'(t)= [1,0,-1;0,2,0;1,0,1]x(t) for a.) x(0)= [-2;2;-1] b.) x(-pi)=[0;1;1]

Let G be an open subset of C ( complex plane) and let P be a polygon in G from a to b. Use the following 2 theorems to show that there is a polygon Q in G from a to b which is composed of line segments which are parallel to either the real or imaginary axes. The 2 theorems are: 1). Theorem: Suppose f: X --> omega is continuou

Show that transformation W (Z) = (a Z + b) / (c Z + d) of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition a d > b c

To find theta, and prove that {e^{in theta} : n nonnegative integer} is a dense subset of the unit circle.

Show that { cis k : k is a non-negative ineger} is dense in T = { z in C ( C here is complex plane) : |z| = 1 }. For which values of theta is { cis ( k*theta) : K is a non-negative integer} dense in T ? P. S. cis k = cos k + i sin k, i here is square root of -1. I want a full justification for each step or claim.

Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p / S = V ( / = intersection). Recall from analytuc geomerty that P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}. Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_3)

What is the solution (-1+i squareroot of 3)exponent 9?

Given f(z), determine f'(z), where it exists and state where f is analytic and where it is not: x + i sin y

Obtain, in polar form all values z^(2/3), z^(3/2) and z^pi for each given z. Obtain, in Cartesian form all values z^i, z^(1-i) 2 + 2i

With a labeled sketch, show the point sets defined by the following: |z|<|z-4| 2≤|z+i|≤5

Evaluate |(2-i)^3 / (1+3i)^2|

(See attached file for full problem description) --- - flow around a corner... - components of velocity... --- (See attached file for full problem description)

See the attachment for the problems. --- - the image of the closed rectangular region... - principal branch of the square root... --- (See attached file for full problem description)

A payoff table is given as ..... a. What choice should be made by the optimistic decision maker? b. What choice should be made by the conservative decision maker? c. What decision should be made under minimax regret? d. If the probabilities of E, F, and G are .2, .5, and .3, respectively, then e. What choice should be ma

Please see the attached file.

See the attachment (77-7) for the problem. and you should also see the examples in section 77 in order to solve it. I also attached section 77 (example1-1 and example1-2).

4. (Separation) Seeking a solution u(x, t) = X(x)T(t) for the given PDE, carry out steps analogous to equations (3)?(6), and derive ODE's analogous to (7a,b). Take the separation constant to be ?K2, as we do in (6). Obtain general solutions of those ODE's (distinguishing any special ic values, as necessary) and use superposition