Complex Analysis

Currently struggling with this task.If you manage to solve this I would be immensively thankful if you could explain what theorems etc are being used to solve this! Thanks :)

Thanks a lot!

Let X be a non-empty set and let E denote the collectionof all unitary subsets of X, that is,E = {{x} ; x $\in$ X}.(a) What is the $\sigma $ -algebra generated by E?(b) If A is the $\sigma $-algebra g...

Should one start by factoring the denominator ? Then using a theorem from complex analysis about the residues at the poles of functions of the form p(z)/q(z)?

Is there a straighforward way to compute I don't think using partial fraction is the right approach. 

Suppose that $T \in L(V,W)$. Prove that if Img$(T)$ is dense in $W$ then $T^*$ is one-to-one, where $X,Y$ are Banach spaces and $T^*$ is the adjoint of $T$.

$\tfrac{\mathrm{d} }{\mathrm{d} W } (\vec{a}W) \odot (\vec{b}W)$I am looking for the solution of this derivative of the Hadamard Product.a and b are vectors of dimension |V|, W is a Matrix with dimens...

Let $D \in \mathbb{C^n}$ be a Reinhardt domain and $f \in \mathcal{O}(D)$. For every $\alpha \in \mathbb{Z}^n$ define where $\zeta \cdot z = (\zeta_1z_1, ..., \zeta_nz_n).$ Then $f_\alpha \in \mathca...

Let $f(z) = z^6 + \cos z$. Apply Rouche’s theorem to find the change in argument of $f(z)$ as $z$ travels once around the circle of radius 2, center zero, in the positive direction.

Use Rouche’s Theorem to show that all roots of $z ^6 + (1 + i)z + 1 = 0$ lines inside the annulus $ \frac{1}{2} \leq |z| \leq \frac{5}{4}$.

Prove that \int_{-\infty}^{\infty}\frac{\cos ax}{x^4+1}dx=\frac{\pi}{2}e^{-\frac{a}{\sqrt{2}}}(\cos \frac{a}{\sqrt{2}}+\sin \frac{a}{\sqrt{2}} ).We shouild use the residue theory in Compex analysis. 

Use contour integrals and residue theory to prove that \int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n}). I need all the details of the proof ASAP.

Let $f: \mathbb{D}\rightarrow \mathbb{D}$ be an analytic complex valued function, where $\mathbb{D}$ is the unit disk. How many of fixed points  $f$ may have in $\mathbb{D}$ ?

Exercise 4.33 from Spivak's Calculus on Manifolds. (Attached).For the definitions and theorems: http://www.strangebeautiful.com/other-texts/spivak-calc-manifolds.pdfThe exercise is on page 118-119. Yo...

For each part of the probem, give a single grammatically correct sentence, using terms that could be understood by someone with very little mathematical experience. (Explain what the problem means in...

Question is in the image

Exercise 4.33 from Spivak's Calculus on Manifolds. (Attached).For the definitions and theorems: http://www.strangebeautiful.com/other-texts/spivak-calc-manifolds.pdfThe exercise is on page 118-119. Yo...

Evaluate \frac{1}{2 \pi i}\int_{|x|=1} \frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}dz,by using residue theory. 

Solve exercise 5 in the attached file.

Show that $\Delta \log (|f(z)|)=0$, where $f(z)$ is an analytic function.

I am stuck on the attched two problems. I have no idea how to approach them. 

Compute the complex integral \oint_{|z-2|=2} \frac{\cos e^z}{z^2-4}dz.