Isolated Singularities

Discuss isolated singularities.

Parallelogram

Please see the attached PDF file. I would prefer a PDF file solution. Thank you! Not that the statement is of the "if and only if" type and so required proof of both directions. I would like someone other than OTA #103642 to provide a solution. Thanks!

Points in a Complex Plane that are the Vertices of a Parallelogram

Find necessary and sufficient conditions (with proofs) such that the points z1, z2, z3, and z4 in the complex plane are the vertices of a parallelogram. I have read that the points z1, z2, z3, and z4 in the complex plane are vertices of a parallelogram if and only if z1 + z3 = z2 + z4. But, if this is indeed the case I would li ...continues

Algebraic series of complex numbers

Given an algebraic series with the following properties: The first term: a1=k-7i The difference: d=-1+2i The sum of the first n terms: S=-5+20i Find k

Pre-calc

Find all the solutions to 5(cos x)^2 - 4cosx-1=0

Find z all solutions

Please see the attached file for full problem description.

z solutions

All z solutions to z^3=9-13i

Complex Logarithm : Proof

Given: z1 = i and z2 = -1+i Show that: Log (z1 z2) ≠ Log z1 + Log z2

Evaluating an Integral using Jordan's Lemma

The problem is to find the value of the integral from 0 to INF of [(ln x)^2]/(x^2 +9). We are to use f(z)= [(Log z)^2]/(z^2 +9), where -pi/2 < Log z < 3pi/2. We are to use the curve C from -R to -p along the real axis, -p to p around 0, p to R along the real axis, and the curve Cr from 0 to pi. I am having several probl ...continues

Evaluating an Integral using Jordan's Lemma

The problem is: Evaluate the integral from 0 to INF of: [(x^(1/3))*(ln x)]/(x^2 +9) dx by using f(z)= [(z^(1/3))*(Log z)]/(z^2 +9), with -pi/2 < Log z < 3pi/2. Also, with z^(1/3)= e^[(1/3)Log z]. We are to use the curve C: from -R to -p, -p to p around origin, p to R, and Cr from 0 to pi. Many thanks in advance ...continues