Use geometric series to find the Laurent series for f (z) = z / (z - 1)(z - 2) in each annulus
(a) Ann(1,0,1)
(b) Ann(1,1,∞)
Ann(a,r,R)
a= center, r=smaller radius, R=larger radius
Ann(1,0,1)=D(1,1)\{0}
My attempt:
f(z)= -1/(z-1) + 2/(z-2)
geometric series: Σ[n=0 to inf] z^n - 1/2...
For the following matrix A, find a unitary matrix U such that U*AU is diagonal:
A =
1 2 2 2
2 1 2 2
2 2 1 2
2 2 2 1
I found the eigenvalues to be -1,-1,-1,7
and the eigenvectors to be (v1)=(-1,1,0,0),(v2)=(-1,0,1,0),(v3)=(-1,0,0,1),(v4)=(1,1,1,1)
Normalize these vectors...
Find all solutions to the following system of linear equations:
(x1) – 2(x2) – (x3)+(x4)=1
2(x1) – 3(x2) + (x3) – (x4)=6
3(x1) – 3(x2) + 6(x3))=15
(x1) + 5(x3)+(x4)=9
Using a system of linear equations, I found:
1 -2 -1 0 1
0 1 3 -3 4
0 0 0 6 0
0 0 0 0 0
so three solutions are...
Not really sure where to start, any help pointing me in the right direction would be great. I've done a somewhat similar problem where I proved that the function was constant, so that f equals the same value at all points, but that is not the case with this function.
Suppose f is an entire function and, for every z in the complex plane, |f'(z) - (2 + 3i)| ≥ 0.00007.
Suppose also that f(0) = 10 + 3i and f'(7+ 9i) = 1 + i. What is f(1 - 4i)?