Question: How to get calculate the integration of follwing:

$$\int\_{0}^{2\pi} \frac{dt}{a + cos t} (a>1)$$

My attempt:

let, $z=e^{it}$
$\implies dt = \frac{dz}{it}$

and 

$$\cos t = \frac{z + \frac{1}{z}}{2}$$

On substituting everything in the integral I got:

$$\frac{2}{i}\int\_{c} \frac{dz}{z^2+2az+1}$$

Now how do I decompose this fraction so that I can use the Residue Theorem? Or is there anyother way to solve this??

Thanks for the help.

Answer: 
$\dots$ is not a mathematical operator. If your prof thinks his proof is correct, then he should be able to write it without $\dots$ . When $\dots$ is used correctly, it is simply tedious to replace it with symbolic logic. In the above case, it is impossible to replace it with symbolic logic so the professors proof is wrong.

Many attempts at proofs of induction can be disregarded due to a simple transform: instead of considering the natural numbers $\mathbb N$, consider the set of polynomials with natural number coefficients, $\mathbb N [x]$. $\mathbb N [x]$ shares most properties of natural numbers, such as being injective (wrt adding one) and having a zero. It is also totally well ordered. However, it is not an inductive set. So any "proof" of induction that doesn't make an assumption true of $\mathbb N$ but not $\mathbb N[x]$ is wrong.
I wouldn't say that he's necessarily wrong, but it's a questionable approach. There **could** be a hidden circular reasoning buried in the details, but basically without the proof of the details you cannot say.

The definition of $\mathbb N$ basically already contains the induction principle in the third requirement. Let $A=\{x\in\mathbb N: \phi(x)\}$ one could easily see that this is a subset of $\mathbb N$ that fulfills the first and second requirements and therefor according to the third requirement we have $A=\mathbb N$, therefore $\phi(x)$ holds for all $x\in\mathbb N$. This shows that basically anything provable with the induction principle can be proved using the third requirement instead (so anything you say is proved by using the induction principle could have been proven without it).

It's not just the $\cdots$ part that's problematic in his proof, also the fact that there's a predecessor chain at all, the fact that it would terminate at $1$ need to be proven first too. Any these proofs that are prerequisite of the proof of the induction principle **could** have used induction principle itself (but could have been shown using the third requirement instead). Even if it's not wrong to do so one would question why one would produce a lot of proofs that begs for the induction principle in order to prove the induction principle when it was readily provable directly instead (it's like a mathematical Rube Goldstein Machine).

Actually you will have to use the third requirement to end up with the induction principle anyway (so you can not use that as an excuse - that you're using a more complicated proof to get rid of one prerequisite).
If you drop the last statement you'll end up in a definition from which you can't prove the induction principle. To see this one could for example construct a set that fulfills the Peano's axiom except the induction principle that it doesn't: for example just extend $\mathbb N$ with a new element $\infty = \Phi(\infty)$. We see that the first two requirements are met, but not the third - in this set we can easily see that induction principle doesn't work as we could for example consider $\phi(x): x\in\mathbb N$.

One detail about that formulation of the induction principle is that it's a statement about an arbitrary statement. Formally this isn't allowed and therefore it should be considered what's called a theorem schema. This means that for each statement $\phi$ you can substitute that into the schema and get a valid theorem instance. 

A proof of such a schema is not the same as a proof in the formal sense as it's more of a recipe of how to prove any theorem instance. As such the proof schema itself need not be formally verifiable as long as each instantiation of it is. Now if there's a concrete counter example we would have a $\phi$ and $k$ and expanding the proof schema would actually result in a formally valid proof.