Complex Analysis

Thanks a lot!

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 $f(z) = z^6 + \cos z$. 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...

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.