Complex Analytic Functions : Cauchy-Riemann Equation

Please see the attached file for the fully formatted problems.

Problems

(See attached file for full problem description) The first part I'm having a problem with bringing (a) greater than or equal to 0 into the picture. Second part, thinking it may be infinite series, although when I tried I had a hard time with convergence

Rearranging Expressions and Residues (Find the two singular points and the residue for each : exp(tz)/(z2 + 2z +17) (t>0))

Find the two singular points and the residue for each : exp(tz)/(z2 + 2z +17) (t>0) Rearranging equations (1st attachment)

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

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

Bounds for analytic functions

If p(z)=a0+a1z+.....+anz^n ia a polynomial and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M. Note:(a0 means a subscript 0, a1z means a subscript 1 times z, anz^n means a subscript n times z to the n power, and ak means a subscript k)

Cauchy principal value, residue

Verify the integral formula with the aid of residues. 1.) Show that the p.v. of the integral of (x^2+1)/(x^4+1) from 0 to infinite = (pi)/(sqrt 2). Note: p.v.=principal value; pi is approximately 3.14; sqrt 2=square root of 2 Please show all work and explain the steps, especially how you found the zeros of the ...continues

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.

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.

Complex Variables : Stereographic Projections, Hyperbolic Functions and Integrals and Circles

Please see the attached file for the fully formatted problems. 1. Let P1 and P2 be two points on the unit sphere x2 + y2 + z2 = 1, and w and w2 the corresponding points on the plane z = 0 under stereographic projection. Show that if P1 and P2 are antipodal points on the sphere, then W1W2 = —1. 2. The hyperbolic functions sin ...continues

Complex Variables : Use definition of limit to prove .... Finding Epsilon and Delta

Use definition of limit to prove .... Please see the attached file for the fully formatted problems.