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Complex Analysis

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

Cardinality for Irrational Numbers

1. Find the cardinality of the set of all irrational numbers, and prove your answer is correct. 2a. Is there a line in the x-y plane such that both coordinates of every point on the line are rational? Prove your answer is correct. 2b. Find the cardinality of the set of all complex numbers, and justify your answer. 3a. W

Electromotive Force : Minimizing Time

For an electric circuit, let V=cos 2pi(t) model the electromotive force in volts at t seconds. find smallest positive value of t where 0<and= to t<and= to 1/2 for the values a) V=0 b) V=.5 c) V=.25

Polynomial Proof

Suppose P is a polynomial with real coefficients and P(a+bi)=0. Prove (a-bi)=0

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

Understanding how to use the Queing Theory - Erlang M/G/s/GD/s/infinity.

How do I solve a formula or equation for the Erlang System M/G/s/GD/s/infinity that predicts resource requirements (how many servers) using the known variables (1) new events per unit of time; (2) average time per event; (3) event time service level (must be resolved by duration); (4) percent of events that must meet that event

Working with complex numbers.

Show that if z0 is an nth root of unity(z0 is not equal to 1) then 1+z0+z0^2+...+z0^(n-1)=0. Hence show that cos(2Pi/1998)+cos(4Pi/1998)+...+cos(2Pi*1997/1998)=-1

Harmonic complex conjugates

Given u = y^3 - 3x^2y, find f(z) = u + iv such that f(z) is analytic. The solution is detailed and well presented.

Finding trigonometric forms.

Find the trigonometric form of the complex number where 0 <= theta < 2pi on the equation: r= 1/2 - [(sqrt(3)/2)i]