## complex analysis – Express \$f(t) = cos(4t)+sin(6t)\$ as trigonometric polynomial.

I’m asked to express the function $$f(t) = cos(4t)+sin(6t)$$ as trigonometric polynomial of the form $$sum_{n=-N}^N c_ne^{inwt}$$. My first approach was to use Eulers formula to arrive at $$f(t)= frac{1}{2}e^{i4t}+frac{1}{2}e^{-i4t}+frac{1}{2i}e^{i6t}-frac{1}{2i}e^{-i6t}$$, but I’m not sure if this is correct and how I would turn this into a series.

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## real analysis – derivative of determinant

I am struggling to understand how to use of the definition of a derivative and definition of a limit to compute the derivative of the determinant function of a 2×2 matrix. It is a subset of $$R^4$$ under Euclidean norm.
Im very new to math analysis.
I don’t know where to start, I know that from the derivative def that the lim as h goes to 0 is $$|((f(x+h)-f(x))/h|$$ = a
for f:R->R

## lexical analysis – What makes a nested function in C so complicated?

Disclaimer: I’m not a compiler expert, but some of my best friends are! 🙂

I do a lot of embedded systems programming where stack (and other memory) is limited. Most of my code is written in gcc, so I thought nested functions could be useful for reducing stack usage. Here’s an example of what I had in mind:

(Though the algorithm isn’t important for this question, assume that `list_traverse` takes a pointer to a list head and a function. The function gets called with each element in turn, returning false when it wants the traversal to stop.)

``````bool list_contains(list_t *head, list_t *item) {
bool found = false;
bool contains_aux(list_t *list) {
if (list == item) {
found = true;
return false;
}
return true;
}
return found;
}
``````

I assumed that the nested `contains_aux()` function would simply be compiled “inside” the `list_contains()` and be able to reference the lexically closed variables (`found` and `item`). But I was wrong.

Instead, looking at the generated code, I see that it installs some sort of trampoline function on the stack and calls that before getting to the `contains_aux()` function.

I guess that makes sense: `contains_aux()` can’t share the same stack frame as `list_contains()` because we’re another function deeper in the stack when its called (via `list_traverse()`).

So is there some way to accomplish this sort of lexical closure efficiently? Or should I just resign myself using un-nested functions and passing a “context” object to `contains_aux()`?

## calculus and analysis – question on correct use of Limit for multivariable function

V 12.1 on windows.

This limit $$lim_{(x rightarrow 0,yrightarrow 0)} frac{x^2-y^2}{x^2+y^2}$$ depends on the direction. So the limit does not exist, or could be written as Maple does it, which is $$-1dots1$$, here is the help from Maple on this:

How can one get Mathematica to give this result? Now Mathematica says the limit is $$1$$. I tried the `Direction` option but not able to make it change its mind.

``````f = (x^2 - y^2)/(x^2 + y^2);
Limit(f, {x -> 0, y -> 0})
(* 1 *)
``````

But we see the limit depends on the direction

`````` Limit(Limit(f, x -> 0), y -> 0)
(* -1 *)

Limit(Limit(f, y -> 0), x -> 0)
(*  1 *)
``````

Here is also Maple to confirm

``````restart;
f:=(x^2-y^2)/(x^2+y^2);
limit(f, (x=0,y=0));
``````

Btw, this is not the only one I found, here is another

``````f = (x^2*y^2)/(x^4 + y^4);
Limit(f, {x -> 0, y -> 0})
(* 0 *)
``````

Maple gives

``````restart;
f:=x^2*y^2/(x^4+y^4);
limit(f,(y=0,x=0))
(* 0 .. 1/2 *)
``````

And another one (this one is from youtube actually, so you can see they also say there the limit does not exist)

``````f = (x^4 - 4 y^2)/(x^2 + 2 y^2);
Limit(f, {x -> 0, y -> 0})
(* 0 *)

restart;
f:=(x^4-4*y^2)/(x^2+2*y^2);
limit(f, (x=0,y=0));
(* -2 .. 0 *)
``````

So I have feeling I am not using Limit in Mathematica correctly, or missing something about its correct use, but do not now know how to correct it. As I said, I tried different `Direction` option.

## real analysis – How do I show this map is open?

For a function $$g: mathbb{S^1} to mathbb{S^1}$$ defined by $$g(cos(theta), sin(theta)) = g(cos(atheta), sin(atheta))$$ where $$a$$ is an integer, how do I show $$g$$ is an open map? I know that the idea is to show that for all open sets $$U$$ of $$mathbb{S^1}, g(U)$$ is open. Any help would be appreciated!

## functional analysis – Can we use the duality notation such that the second variable is an element of the measure space?

I read article New Sequential Compactness Results for Spaces of Scalarly Integrable Functions by Erik J. Balder, on page 8 : Author defined the function $$a: Ttimes Eto mathbb{R}$$ by the usual duality between $$E$$ and $$E^*=F$$ $$( ⟨.,.⟩)$$ such that:
$$a(t,x)=langle x, trangle$$
But normally the second variable $$(tin T)$$ must be an element of dual of \$E£, right? Why does the author make this notation?

## fa.functional analysis – Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of \$L^2\$ of the line

Let $$newcommand{ptp}{widehat{otimes}}ptp$$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, suggest to me that the following should be true:
$$newcommand{Real}{{bf R}}$$

for some $$k > 1$$, every compactly supported $$C^k$$-function $$Real^2toReal$$ belongs to $$L^2(Real)ptp L^2(Real)$$.

(Recall that in contrast, there are continuous functions on $$(0,1)^2$$ that do not belong to $$C(0,1)ptp C(0,1)$$.)

If the claim above is true, I would like to know if there are standard references, perhaps from the world of integral kernel operators or Sobolev spaces, which I could cite, rather than reinventing the wheel (and probably getting suboptimal values of $$k$$).

In a slightly different direction, I would also be interested to know of references which prove analogous resuts for $$C^k$$ functions (suitably interpreted) on compact connected Lie groups.

## real analysis – we want to prove that \$(n,n+1) cap mathbb{N}\$ is empty

Indeed, let $$S(n)$$ be the statement $${ n : (n,n+1) cap mathbb{N} = varnothing }$$. Clearly, $$(1,2) cap mathbb{N} = varnothing$$, thus $$1 in S(n)$$. Assume now that $$S(n)$$ is true. Then,

$$(n+1,n+2) cap mathbb{N} = (n,n+1) cap (mathbb{N} setminus (n+1,n+2) ) cap mathbb{N} = (n,n+1) cap mathbb{N} cap ( mathbb{N} setminus (n+1,n+2) ) = varnothing$$

and the claim follows by the principle of mathematical induction.

Is this correct proof?

## History of Microlocal analysis

Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I’m trying to find out how this branch of analysis was developed…