Homework Statement
I have some past exam questions that I am confused with
Homework Equations
a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz
The Attempt at a Solution
I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:
a) it says f(z)...
It's actually
\oint \frac{\cos(z+11)}{z^2-23} dz
and since z = ±√23 is not in the unit circle, it must be 0 right?
The second integral is
\oint \frac{\cos(z+11)}{z^2-6z} dz
which can be written as
\oint \frac{\cos(z+11)}{z(z-6)} dz
then
\oint \frac{\frac{\cos(z+11)}{z-6}}{z} dz
so...
Homework Statement
I just wrote a test and was wondering if I got these questions right, I already solved them, please see the attached pictures below. Here are the questions; sorry for non-latex form
1) Let gamma be a positively oriented unit circle (|z|=1) in C
solve: i) integral of...
Looks like I had a brain fart there haha, jeez I feel dumb. Thanks.
Also, I have another question if you could answer it: what makes a Jordan curve unique?
For example, there is this question: Sketch infinitely many non-homotopic Jordan curves in C\<-1, i, 1>.
I have these two...
This isn't homework but I'm having trouble understanding the concept of non-homotopic and homotopic Jordan curves.
My understanding of Jordan curves and homotopy:
A Jordan curve is a simple closed curve (ie a closed curve that only intersects at the endpoints; f(z1)=f(z2) => z1=z2) such that...
Homework Statement
prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R
Homework Equations
The Attempt at a Solution
is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between...
Rather than dealing with all the terms separately, I've been trying to just apply a simple comparison test (followed by an integral test) to prove its convergence, but I could only come up with divergent cases.
I'm starting to think if there is an error with this question; does this series...
Homework Statement
Let f_{n}(x)=\frac{-x^2+2x-2x/n+n-1+2/n-1/n^2}{(n ln(n))^2}
Prove f(x) = \sum^{\infty}_{n=1} f_{n}(x) is well defined and continuous on the interval [0,1].
Homework Equations
In a complete normed space, if \sum x_{k}converges absolutely, then it converges.The Attempt at...
\alpha = <1, \sqrt{3}(2x-1)> is an orthonormal basis for P_1{}(R)
By the definition of the inner product space, have
< T(p(x)), q(x) > = \int_0^1 \! p'(x)q(x) \, \mathrm{d} x + \int_0^1 \! p(x)q(x) \, \mathrm{d} x
So I computed
< T(1), 1 > = 1
< T(1), \sqrt{3}(2x-1) > = 0
<...
I just got the new matrices with the orthonormal basis but I'm still at the same sticking point
The matrices aren't equal so T isn't self-adjoint; that's all I can conclude right now
[Linear Algebra] Finding T* adjoint of a linear operator
Homework Statement
Consider P_1{}(R), the vector space of real linear polynomials, with inner product
< p(x), q(x) > = \int_0^1 \! p(x)q(x) \, \mathrm{d} x
Let T: P_1{}(R) \rightarrow P_1{}(R) be defined by T(p(x)) = p'(x) +...